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Properties of Fossil Groups of Galaxies

We review the formation and evolution of fossil groups and clusters from both the theoretical and the observational points of view. In the optical band, these systems are dominated by the light of the central galaxy. They were interpreted as old systems that had enough time to merge all the M* galaxies within the central one. During the last two decades many observational studies were performed to prove the old and relaxed state of fossil systems. The majority of these studies, that spans a wide range of topics including halos global scaling relations, dynamical substructures, stellar populations, and galaxy luminosity functions, seem to challenge this scenario. The general picture that can be obtained by reviewing all the observational works is that the fossil state could be transitional. Indeed, the formation of the large magnitude gap observed in fossil systems could be related to internal processes rather than an old formation.


Introduction
The Lambda cold dark matter scenario (ΛCDM) predicts that structures in the Universe form following a hierarchical evolution: small objects collapsed first under their self-gravity and are then merged continuously to build larger structures. In this scenario, galaxy formed first, then merged in small groups and the process continue until the creation of massive galaxy clusters (White & Rees, 1978;White & Frenk, 1991). Ponman & Bertram (1993) firstly suggested, while studying compact groups, that this building scenario could be taken to the extreme consequences. They predicted that, in some cases, all the main galaxies of a group could merge with one another, creating a giant galaxy embedded in an X-ray halo typical of a group. This prediction was supported, one year later, by the discovery of RX 11340.6+4018, an apparently isolated elliptical galaxy, at redshift z = 0.171, found in an extended X-ray halo. The estimated X-ray mass was 2.8 × 10 13 M , making it a typical group-sized object (Ponman et al., 1994). These systems were named as "fossil groups" (FGs). They were thought to be the latest stage in the evolution of galaxy groups. For this reason they were supposed to be old and dynamically relaxed systems (Ponman & Bertram, 1993).
The spatial density and fraction of FGs were estimated in different studies, both theoretically and observationally. The agreement on the spatial density is quite good, with estimations in the range ∼ 1 − 3 × 10 −6 h 3 50 Mpc −3 (see e.g. Vikhlinin et al., 1999;Santos et al., 2007;La Barbera et al., 2009). On the other hand, the fraction of FGs with respect to the total amount of clusters and groups is more debated. A comparison between different works can be found in table 1 of Dariush et al. (2007): the estimated fractions varies between 1% and 40%, with a large scatter, and a mean value can be found at about ∼ 10 − 15%.
In this review we show the main observational and theoretical works related to FGs written during the last three decades. Our aim is to show that these systems fix well in the current theory of structure formation in the Universe. They are just extreme systems produced following this theory. This review is structured in the following way: in Sect. 2 we will describe the search for FGs in the last ∼ 20 years and their observational definitions. In Sect. 3 we will present the theoretical scenarios proposed to describe these systems. In Sect. 4 we will discuss the properties of the intra-cluster medium and of the galaxy populations. Then, we will describe the possible progenitors of FGs in Sect. 5. Finally, in Sect. 6 we will propose a sample of "genuine" FGs and draw our conclusions.

The search for fossil systems
Since their discovery in 1994, a strong observational effort was done to find FGs. Despite their supposed frequency, more than 10 years were needed before producing reasonably large catalogues of some tens of candidates. In this sense, one of the main obstacles for the community was to agree on a practical definition. In fact, at the beginning the search was limited to isolated galaxies surrounded by an X-ray halo, but this definition was too loose (Vikhlinin et al., 1999). More rigorous definitions, involving photometry, spectrocopy and X-ray data, were proposed and are actually used. We will discuss them in Sect. 2.1. However, these rigorous definitions had important observational limitations. The result of these constraints was that fossil samples were not selected in a homogeneous way and, in Sect. 2.2, we will discuss the strength and weaknesses of the different approaches adopted in the literature.

Operational definitions of fossil groups
The most common operational definitions of FGs are based on the magnitude gap between the brightest member galaxy of the group/cluster and other galaxy sorted by their magnitude (∆m 1,j , where j represent the j−th ranked galaxy). In particular, Jones et al. (2003) suggested that a group or a cluster of galaxies should be classified as fossil if the magnitude gap between its two brightest members (∆m 12 ) is larger than 2 magnitudes in the r−band. In addition, they also imposed that the two brightest galaxies should be located within half the (projected) virial radius (defined as r 200 , the radius of a sphere which mean density is 200 times the critical density of the Universe). Another common definition is the one presented in Dariush et al. (2010): a group/cluster of galaxies is classified as fossil if the magnitude gap between its first and fourth brightest member galaxies (∆m 14 ) is larger than 2.5 magnitudes in the r−band and within half the (projected) virial radius. Both definitions also require the presence of a diffuse X-ray halo, with L X ≥ 10 42 h −2 50 erg s −1 . This last criterium ensures that the system is located in a potential well similar in mass to groups or clusters of galaxies. An example of a typical FG and the color-magnitude diagram of its galaxy population is shown in Fig. 1.
The original name and definition were though to describe only galaxy groups. However, many studies (e.g. Sun et al., 2004;Khosroshahi et al., 2006;Aguerri et al., 2011) found the existence of fossil clusters. This is due to the lack of an upper limit for the X-ray luminosity in both the most-common operational definitions. As a consequence, when describing the general properties of the population, fossil groups and fossil clusters are interchangeable in the literature. Besides, a more general fossil systems is also widely used. We invite the reader to consider these three definitions as equivalent along this review.

Catalogues
We already mentioned that the prototype of FGs is RX 11340.6+4018, presented in Ponman et al. (1994). A few years later, Vikhlinin et al. (1999) studied a sample of four X-Ray Overluminous Elliptical Galaxies: the authors claimed that these objects could have been part of the fossil category, but it was just a suggestion, since no operational definition was available until Jones et al. (2003) proposed the use of the magnitude gap as the discriminating factor between fossils and non-fossils, proposing a sample of 5 FGs in their work.
Other studies presented small numbers of FGs candidates (Yoshioka et al., 2004;Khosroshahi et al., 2004;Sun et al., 2004;Ulmer et al., 2005;Mendes de Oliveira et al., 2006;Khosroshahi et al., 2006, amongst others), whereas the first large sample of FG candidates was presented in Santos et al. (2007): in this work, the authors selected 34 galaxy aggregations obtained from the Sloan Digital Sky Survey Data Release 5 (SDSS DR5, Adelman-McCarthy et al., 2007) by cross matching the sample of SDSS Luminous Red Galaxies (Eisenstein et al., 2001) with sources in the ROSAT all-sky bright source catalogue (Voges et al., 1999). The result of the cross match was a list of elliptical galaxies surrounded by an X-ray halo. Then, the magnitude gap was computed within a fixed radius (500 kpc) and in a fixed redshift range (∆z = 0.002 when spectroscopic redshift is available, ∆z = 0.035 when only photometric redshift is known). These 34 FG candidates were then studied in detail by the Fossil Group Origins (FOGO) project (Aguerri et al., 2011). This observational project produced a set of results analysing the properties of these systems. In the framework of this project, Zarattini et al. (2014) confirmed that 15 +8 −5 of the candidates are FGs according to the Jones et al. (2003) or Dariush et al. (2010) definitions. The uncertainties in the number of confirmed FGs reflect those on the definition of r 200 . The large difference between the proposed candidates and the confirmed fossils can be explained with a stricter implementation of the definition criteria (e.g. differences in the search radius and membership definition). From the comparison between Santos et al. (2007) and Zarattini et al. (2014) it seems clear that three types of observations are needed in order to strictly define an FG: (i) X-ray data, required to estimate the mass of the system and define the virial radius, (ii) multi-object  2011) semi-analytic model. Data point are colour-coded according to the ratio of the group halo mass at redshift z ∼ 1 to its mass at z = 0. The plane has been sub-divided into blocks within which the probability that the halo is old or young is given. In this diagram, panels (5), (9) and (10) contain mostly old systems while the panels (3), (4) and(8) are mostly occupied by young systems. The image is taken from Raouf et al. (2014), in particular it corresponds to their fig. 1. spectroscopy, in order to identify the real members of the FG, removing fore-and background objects, and (iii) an optical image, needed to measure the magnitude of each galaxy and to compute the magnitude gap between the brightest galaxy and the other members. The combination of the strong observational effort together with the demanding observational definition is probably the main limit for the building of a large and homogeneous dataset of fossil systems. However, the number of known FGs kept growing in the last decade. Without intending to be exhaustive, La Barbera et al. (2009) used SDSS-DR4 and ROSAT to build a sample of 25 FG candidates, Tavasoli et al. (2011) found 109 FGs obtained from SDSS-DR7 using a friend-of-friend algorithm, Makarov & Karachentsev (2011) presented a catalogue of 395 nearby groups and claimed that ∼ 25% of those are FGs, Harrison et al. (2012) presented a sample of 17 FG candidates, and Gozaliasl et al. (2014) presented a catalogue of 129 groups, of which 22 ±6% are fossils. While the number of candidates is rising, it become more and more complicated to have dedicated multi-object spectroscopic observations to constraint membership. The main drawback of this strategy is that the purity of the sample is difficult to control and non-fossil systems can dilute the statistical relevance of the results. It is thus complicated to estimate the number of "genuine" FGs, namely those that strictly accomplish the operational definitions, known up to date. However, in the last section of this review we will try to define a sample of such genuine systems.

Theoretical framework
Numerical simulations of the number of satellites in galaxy groups predict the presence of a huge number of dwarf galaxies (e.g. Moore et al., 1999). However, these predictions are not confirmed in observations. In fact, if the number of bright galaxies is in agreement with these predictions, the number of observed dwarfs is up to two orders of magnitude lower than expected (e.g. Mateo, 1998). This is the so-called missing satellite problem. D' Onghia & Lake (2004) pointed out that FGs could scale up the missing satellite problem at the mass scales of more massive galaxies. In particular, they compared the number of satellites predicted by ΛCDM at different halo mass scales and concluded that FGs shown smaller number of galaxies such as the Milky Way and the Large Magallanic Cloud than those predicted by the structure formation theory. They claimed that the reason of this lack of bright galaxies was due to over merging processes occurred in FGs. These early findings made FGs extreme objects in the structure formation of the Universe. The over merging processes in FGs could be explained in terms of differences in the orbital structure between fossil and non-fossil systems. Sommer-Larsen (2006) used cosmological TreeSPH simulations to study the orbital structure of the intra-group (IG) stars for a set of clusters with masses ∼ 10 14 M . He concluded that the velocity distribution of the IG stars was significantly more radially anisotropic for fossil than for non-fossil systems. This pointed out that the initial velocity distribution of the group galaxies could play an important role in defining the fossil status of the system.
Several works have focused on the theoretical study of the mass assembly history of fossil and non-fossil systems. One of the first papers on this topic was done by D' Onghia et al. (2005a) where they analysed the mass assembly of systems with M vir ∼ 10 14 M . They found a correlation between ∆m 12 and the formation time of the group defined as the redshift at which 50% of the total mass of the system at z = 0 is already in place (z 50 ). In particular, they found that FGs have assembled more than half of their present mass at z > 1, with a subsequent growth by minor mergers alone. This early assembly gave to FGs enough time to merge their M * galaxies (where M * is the characteristic magnitude of the luminosity function, see Sect. 4.2.3 for details) and produce the large magnitude gaps observed at z = 0. The mass assembly of the low-mass group regime (M vir ∼ 10 13 − 10 13.5 M ) was analysed by using the Millenium Simulation by Dariush et al. (2007Dariush et al. ( , 2010. In particular, these authors found that the selection of the systems using their magnitude gap alone does not guarantee the selection of early formed systems. They observed that the majority of the objects that have assembled more than 50% of their halo mass at z = 1 are not fossil systems today. A similar result was found in Deason et al. (2013): 20% of the groups selected from the Millenium Simulation with large mass gap (similar to fossil systems) turned to be young objects. Raouf et al. (2014) proposed that a combination of three observational parameters (magnitude gap, luminosity of the brightest cluster galaxy and its offset from the group luminosity centroid) considerably improve the selected rate of dynamically old systems. In Fig. 2 it can be seen that the probability for a system with ∆m 12 > 2.0 and ∆m 14 > 2.5 to be old grows when the absolute magnitude of the BCG is smaller. In this case, Raouf et al. (2014) defined a group as old if its halo has over 50% of its final mass at z = 1 and young if this fraction is less than 30%. Indeed, this plots demonstrate that ∆m 14 works better than ∆m 12 , when combined with the absolute magnitude of the central galaxies, in finding old systems. This result was recently confirmed in Zhoolideh Haghighi et al. (2020), since the authors found a clear correlation between ∆m 12 , the offset of the luminosity centroid, and the dynamical age of a group/cluster. However, this criterium is less used in the literature, probably because it is newer than the Jones et al. (2003) and Dariush et al. (2010) ones.
Also the Illustris Simulation was used to analyse the properties of the FGs in the mass regime 10 13 − 10 13.5 M (Kundert et al., 2017). The authors found that the magnitude gap of FGs identified at z = 0 were on average created about 3 Gyr ago. In addition, the fossil central galaxies became more massive than non-fossil ones. This difference was explained as due to differences in the mass acquired through mergers between z = 0.1 − 1, as can be seen in the left panel of Fig. 3. Fossil BCGs have also a larger number of major mergers than non-fossil ones (right panel of Fig. 3). Indeed, the last major merger of fossil BCGs was later (e.g. at lower redshift). No differences were found in the distribution of the time formation (z 50 ) of fossil and non-fossil halos. The group mass assembly of fossils and non-fossils differs only in the recent group accretion history, in particular in the formation time of the 80% of the mass of the halo. However, semi-analytical models studied the dynamical evolution of galaxies in groups with different formation epochs (Raouf et al., 2018). They found that BCGs of dynamically-young groups suffered the last major galaxy merger ∼ 2 Gyr more recently than their counterparts in dynamically old groups and that FGs are somewhere in the middle between the other two populations. However, these authors found a lack of recent major mergers in FGs, that is in agreement with the evolution of their old systems.  Raouf et al. (2016) also used the Illustris Simulation to study some properties of fossil systems. They found that this simulation overproduce FGs in comparison with observations and semi-analytical predictions. They also obtained that the intra-group medium (IGM) in dynamically evolved groups is hotter, for a given halo mass, than that in still evolving ones.
Several studies found that the fossil phase of a system could be transitional. Galaxy clusters and groups pass through fossil and non-fossil phases along their evolution. von Benda-Beckmann et al. (2008) found a population of groups that presented a fossil phase at high redshift which is terminated later by the accretion of new bright galaxies. The transitional phase of the fossil status is also reported by other simulations like Kundert et al. (2017). This fluctuations in the magnitude gap could be related with the large-scale environment in which the systems are located. Indeed, Diaz-Gimenez et al. (2011) found that, in the Millenium Simulation, the environment was different for fossil and non-fossil systems with similar masses. They showed an increase in the local density profile of galaxies at ∼ 2.5 r vir from the group centers. This increment was more noticeable in fossil than in non-fossil systems and was linked with the earlier formation time of fossil groups. We will discuss in Sect. 5 what is found in observations that can be linked to the transitional fossil phase.
The properties of the galaxy populations in fossil and non-fossil systems have been also analysed using cosmological simulations. Romeo et al. (2015) reported from cosmological hydrodynamical simulations and semi-analytical models that fossil and non-fossil systems show different star forming rates at low z, being indistinguishable at z > 0.5. In contrast, Kundert et al. (2017) found no differences in the stellar age, metallicity and star formation rates of BCGs in fossil and non-fossil systems from the Illustris Simulation. Raouf et al. (2016) analysed the properties of the black holes developed in the center of the BCGs for fossil and non-fossil systems. They found that the mass of the black holes hosted in BCGs is larger in dynamically evolved groups with a lower rate of mass accretion, a result confirmed also in Khosroshahi et al. (2017). Kanagusuku et al. (2016) found that fossil and non-fossil systems selected from the Millennium Simulation shown different galaxy populations. In particular, at early times, FGs comprised two large brightest galaxies surrounded by faint ones. At the faint end of the luminosity function, fossil systems turned to be denser at early times that non-fossil ones. This trend reverses at later time and became similar before z = 0. This was caused by an increase at a constant rate of the number of faint objects in non-fossil systems. In contrast, the number of faint galaxies reached a plateau at z ∼ 0.6 in FGs, and then grows faster towards z = 0. The evolution of the galaxy luminosity function as a function of redshift for fossil and non-fossil systems was studied by Gozaliasl et al. (2014). They build up luminosity functions of galaxy aggregations based on the Millennium Simulation. They found that the bright end of the galaxy luminosity function strongly evolved for fossil systems from z = 0.5 to z = 0, with changes in M * ∼ 1.2 mag. This suggest that the mergers of the M * galaxies in fossil systems have a significant impact in the formation of the bright cluster galaxies. In contrast, the faint-end slope of the luminosity function shows no considerable redshift evolution in fossil systems, unlike in non-fossil ones where it grows by 25 -42 % towards low redshifts.

Observational properties
In this section we review the observational properties of FGs. In Sect. 4.1 we describe the properties of the intra-cluster medium, in particular we present global scaling relations, mass and entropy profiles, cool cores, halo concentrations, and metallicities. In Sect. 4.2 we analyse the galaxy population and, in particular, FGs' luminosity functions, galaxy substructures, stellar populations, central galaxies, and large scale structures.

Properties of the halos
The intra-cluster medium (ICM) is the largest baryonic component in galaxy clusters, responsible for ∼ 10% of the total mass of the cluster (e.g. Rosati et al., 2002). In fact, galaxy formation is inefficient and only ∼ 10% of the gas is converted in stars and galaxies (Voit, 2005), leaving the vast majority adrift in the intra-cluster space. This gas is trapped in the deep potential well of the cluster and heated to X-ray-emitting temperatures through shocks and adiabatic compressions (Ghirardini et al., 2019).
Gravitational collapse predicts tight scaling relations between ICM and cluster mass, according to the so-called self-similar model (Kaiser, 1986;Bryan & Norman, 1998). Moreover, cosmological simulations predict that the scaled thermodynamical profiles of galaxy clusters are nearly universal (e.g. Frenk et al., 1999). For these reasons, the ICM is a powerful tool to study the formation and evolution of galaxy clusters: deviation from the gravitational collapse predictions can be used to investigate non-gravitational physics, such as cooling and feedback from supernovae and active galactic nuclei (AGN).

Global scaling relations
In the left panel of Fig. 4 we show the correlation between X-ray temperatures and luminosities for various samples of FGs and non-FGs. It can be seen that both types of objects are found in the same, tight, correlation. We won't enter in a detailed discussion of purely X-ray relations and their meaning, we refer the reader to the companion review by Lovisari et al. (2021) for a description of X-ray scaling relations in galaxy groups. The result presented in the left panel of Fig. 4 can be extended to all those relations that only involve X-ray data: FGs and non-FGs are usually found in the same correlations. This seems to indicate that FGs and non-FGs are formed in a similar way.
However, discussion arose in those studies focused on the comparison between X-ray and optical properties of FGs and non-FGs. In fact, Jones et al. (2003) and Khosroshahi et al. (2007) found FGs to be over luminous in X-rays using samples of five and seven FGs, respectively. They claimed FGs to be a factor 5-10 brighter than regular groups or clusters in the X-ray.
On the other hand, more recent studied found no differences in the statistical properties of FGs and non-FGs samples (Voevodkin et al., 2010;Harrison et al., 2012;Girardi et al., 2014). In particular, these authors claimed that it is crucial to use homogeneous data and procedures to analyse both the FG and the control samples. According to their statements, this could be the source of the excess of X-ray luminosity (or the lack of optical luminosity) found in the previous studies.
Nevertheless, Khosroshahi et al. (2014) discussed a sample of groups, one of which defined as fossil, that lies above the L X − L opt relation of non-fossil systems, reopening the debate on fossil system scaling relations.
The (possible) final point of this debate was put by Kundert et al. (2015): as well as demonstrating that no differences are found between fossils and non-fossils in their sample of 10 groups and clusters observed with specific Suzaku follow up, they recomputed the luminosities of the FGs from Khosroshahi et al. (2007) and Proctor et al. (2011) in a homogeneous way. In the right panel of Fig. 4 we show the relation presented in Kundert et al. (2015). The authors concluded that the discrepancies in the literature can be reconciled if X-ray and optical luminosities were computed using the same bands and radii, pointing out that many differences could be due not only to the low statistics but also the lack of homogeneous datasets.
In parallel with the discussion on the L X − L opt relation, a debate on the mass-to-light (M/L) ratio of FGs arose. In fact, if there is a chance that FGs are under luminous in the optical bands, they should have larger M/L ratios than non-FGs . Again, the studies available in the literature are somewhat contradictory. Sun et al. (2004); Khosroshahi et al. (2004Khosroshahi et al. ( , 2006 found normal values for their samples of FGs, compatible with non-fossil systems, although if marginally darker for a fixed mass. On the other hand, Vikhlinin et al. (1999) found a high M/L for their sample of overluminous elliptical galaxies (OLEGs, about 3 times larger). The same result was found by Proctor et al. (2011) in analysing a sample of 10 FG candidates. The authors suggested that FGs are simply dark clusters: they are characterised by a mass and a central galaxy that are typical of galaxy clusters, but embedded in a poor environment, so that the richness and the total optical luminosity are below the non-fossil ones. Finally, Yoshioka et al. (2004) also found high M/L ratios for their sample of "isolated X-ray overluminous elliptical galaxies". However, we suggest that inconsistencies in the measured M/L ratios could be due to differences in the methodology and quality of the data. For example, the optical luminosities of Vikhlinin et al. (1999) and Khosroshahi et al. (2006) are computed in the R band and within r 200 for every object in the sample, those of Proctor et al. (2011) in a variable radius of 500 − 1000 kpc, whereas Sun et al. (2004) estimated a normal M/L ratio from the gas fraction profile out to 450 kpc, and Yoshioka et al. (2004) uses the B-band luminosity of the BCG as the total luminosity of the system. Thus, it seems reasonable that the disagreement between different results could be healed only with a homogeneous study of a large sample of FGs. As we already mentioned for other topics, such a study is far from being performed.

Mass and entropy profiles
The study of mass profiles in clusters is an important tool to confirm the ΛCDM paradigm. In fact, this model predicts a universal mass profile that does not depend on the mass of the cluster and it is usually assumed to have the shape of a Navarro-Frenk-White profile (NFW, Navarro et al., 1997).
Mass profiles were studied mainly for individual FGs, or small samples, making difficult to extrapolate general conclusions. Possibly the first mass profiles of FGs to be computed were those of NGC 6482 (Khosroshahi et al., 2004), RX J1416.4+2315 (Khosroshahi et al., 2006), and ESO 3060170 (Sun et al., 2004). The first two systems shows a mass profile well described by an NFW, with a high central concentration that was interpreted as a sign of early formation. On the other hand, the mass profile of ESO 3060170 showed a flattening in the external regions not compatible with numerical simulations and also confirmed in Su et al. (2013) out to the virial radius. Yoshioka et al. (2004) studied the mass profiles of 4 FG candidates, finding no differences with normal groups/clusters. Also Gastaldello et al. (2007) studied the mass profile of ESO 3060170, within a sample of 16 relaxed groups and clusters: they found a good agreement with a NFW profile, but it must be noticed that their data reached R ∼ 200 kpc, whereas Sun et al. (2004) and Su et al. (2013) data reached R ∼ 500 kpc and R ∼ 1000 kpc, respectively. The three mass profiles are comparable within R ∼ 200 kpc and differences with the standard NFW profiles rose in the most external regions. This clarify again the complexity of the comparison when individual FGs are studied using data from different sources, within different radius, and treated with different techniques. Another two mass profiles for the FGs RXC J0216.7-4749 and RXC J2315.7-0222 were studied in Démoclès et al. (2010). Only the latter has a good profile, well fitted, out to R 500 , by a NFW profile plus a central stellar component.
Entropy is also of great interest because it controls ICM global properties and records the thermal history of a cluster, since it is conserved in adiabatic processes. Entropy is therefore a useful quantity for studying the effects of feedback on the cluster environment and investigating any breakdown of cluster self-similarity. Most of the studies cited for the discussion of the mass profiles were also able to compute an entropy profile. Again, since only individual systems were studied, the results are somewhat controversial and it is not trivial to generalise the conclusions to the entire FG category. In particular, Démoclès et al. (2010) found that the entropy profiles out to R 500 of their two FGs show a considerable excess above the expectations from non-radiative simulations, especially for RXC J0216.7-4749. This is expected if significant non-gravitational processes affect the ICM. Su et al. (2013) found and entropy profile that is in agreement with simulations out to ∼ 0.9 R 200 and then flattens in the outskirts, due to gas clumpiness and outward redistribution. Humphrey et al. (2012) studied RXJ 1159+5531 combining Suzaku, Chandra, and XMM observations to find no evidence of the flattening in the entropy profile outside ∼ R 500 . A similar results was also found in Su et al. (2015) for the same cluster: its entropy profile is consistent with predictions from gravity-only simulations.
The urge for a systematic study of a large sample of FGs in the X-rays appears as necessary to constraint the average mass and entropy profiles of these objects. However, it seems difficult to realise, due to the small number of nearby FGs that can be deeply observed with current X-ray facilities. An improvement on this side is expected with the next all-sky survey that will be taken by eROSITA and Athena missions. The former is expected to find ∼ 10 5 X-ray clusters and groups, the latter will take advantage of its high-resolution to better constraint radial profiles of, for example, temperature, density, and mass.

Cool cores
Early observations of the gas in galaxy clusters found that it was so dense in the central regions that its cooling time was much shorter than the Hubble time (e.g. Lea et al., 1973). The majority of the clusters studied in the literature show these cool cores (CC, e.g. Vikhlinin et al., 2005;Hudson et al., 2010). CCs are usually associated to relaxed clusters, since mergers easily erase them. For this reason, it appeared as natural to look for CCs in FGs, in order to confirm their old and dynamically relaxed status.
The first study on CCs for a sample of FG candidates was done by Vikhlinin et al. (1999). In their work, the authors studied four isolated elliptical galaxies selected from ROSAT X-ray data and confirmed as fossil systems with a dedicated optical follow up. It is worth noting that the authors suggested that these objects could be FGs, but at the time no operational definition was available, so no ∆m 12 is computed in their paper. However, at least three out of four were later confirmed as FGs in other publications, confirming the accuracy of their approach. Vikhlinin et al. (1999) results indicated the presence of CCs in the central regions of these objects.
Later works were focused on deeper studies of individual FGs and found controversial results: Khosroshahi et al. (2004Khosroshahi et al. ( , 2006 found no central drop in NGC 6482 and RX J1416.4+2315. On the other hand, Sun et al. (2004) found a CC in ESO 3060170, as well as Démoclès et al. (2010) for RXC J0216.7-4749 and RXC J2315.7-0222 and Su et al. (2015) for RXJ1159+5531. It is interesting to note that Miraghaei et al. (2014) studied three of the cited clusters (NGC 6482, RX J1416.4+2315, and ESO 3060170) using radio observations, finding signs of recent AGN activity only in the first two. However, the AGN power computed was not sufficient to remove CCs from these clusters.
The need for larger samples was partly satisfied only recently, when Bharadwaj et al. (2016) studied a sample of 17 FGs for which Chandra archival data were available. They defined three different diagnostic to evaluate the presence of CCs and they found that ∼ 80% of FGs showed clear hints of the presence of CCs (e.g. at least two diagnostics compatible with the CC).
It seems, thus, reasonable to claim that FGs are mostly cool-cored. However, the fraction of FGs with a CC is similar to that of non-FGs. For example, Hudson et al. (2010) studied a large sample of 64 galaxy clusters for which high-quality X-ray data from Chandra were available, finding that ∼ 70% of their clusters host a CC. It thus seems that this is not a peculiar behaviour of FGs.

Halo concentration
One of the main parameters of the NFW model is the concentration, usually computed as c ∆ = r ∆ /r s , where r ∆ represent the radius of a sphere of mean interior density ρ ∆ and r s is the scale radius of the NFW profile. Typical values of ∆ are 200 (often assumed to be equivalent to the virial radius) or 500. Navarro et al. (1997) pointed out that the concentration parameter reflects the density of the Universe when the halo formed. In particular, older halos formed in higher-density environments and tend to have larger concentrations. Several theoretical studies found that FGs assembled half of their mass at earlier epochs than non-fossil ones (see e.g. D' Onghia et al., 2005b;von Benda-Beckmann et al., 2008). In this framework, it is expected that FGs would be located in high concentrated halos. Different numerical models are used in the literature to compute halo concentration in clusters (e.g. Dolag et al., 2004;De Boni et al., 2013). Thus, observational results can be easily compared with theoretical predictions to test the formation scenario of FGs.
From an observational point of view, various methods can be used to compute the c parameter. A first approach to derive the central concentration of DM halos is to measure the gas mass profile in the X-rays out to r 200 or r 500 , fit it with an NFW profile and then use the derived r s to compute the concentration. Using this approach, Khosroshahi et al. (2004) Khosroshahi et al. (2006) Khosroshahi et al. (2007), and Buote (2017) found concentration values higher than expected for their samples of individual FGs Other authors, like Démoclès et al. (2010) and Pratt et al. (2016), found normal concentrations in a total of 6 FGs. Again, comparing results with such small statistics and taking into account non-homogeneities in the analytic procedures makes difficult to reach a final conclusion that can be applied to the mean FG population.
However, other approaches can be used to measure halo concentration. In particular, Vitorelli et al. (2018) stacked ∼ 1000 systems from the CS82 survey (Moraes et al., 2014) in different magnitude-gap bins, the larger of which has mean ∆m 12 ∼ 1.7. They cross-correlate weak lensing measurements with NFW parametric mass profiles to measure masses and concentrations of their sample. They found that halos in the ∆m 12 ∼ 1.7 bin have a higher probability to be more concentrated and, thus, probably formed earlier.
Finally, the halo concentration can be estimated using the velocity of member galaxies as tracers of the underlying mass distribution. This was done in Zarattini et al. (2019), were the authors analysed a sample of ∼ 100 clusters and groups, dividing them in different magnitude-gap bins. For each bin, they stacked all the available galaxies to increase the statistic. They found c 200 = 2.5 ± 0.4 for the bin with the largest magnitude gap (defined as ∆m 12 > 1.5). This values are in agreement, within the uncertainties, with their results in the other three magnitude-gap bins, as well as with other similar work in the literature of non-fossil clusters (e.g. Lin et al., 2004;van der Burg et al., 2015, and references therein). However, the large uncertainties typical of this observational technique prevented the authors from reaching a strong conclusion.

Metallicity
The hot intra-group gas contains elements that are typically synthesized in stars and SNe. For general details on this topic we refer the reader to the companion review of Gastaldello et al. (2021). Here, we focus on the single study conducted on metal abundances in FGs, presented in Sato et al. (2010). The authors get Suzaku data out to 0.5 r 180 for NCG 1550 and were able to confirm that the abundance ratios O/Fe, Mg/Fe, Si/Fe, and S/Fe are similar to those of other poor groups observed with the same satellite. Moreover, the number ratio of type-I and type-II SNe computed in Sato et al. (2010) is also similar to that obtained for non-fossil groups. As a consequence, their work can be included in those that are not finding differences between FGs and non-FGs.

Galaxy population
The study of the galaxy population in FGs is mainly done in the optical range. In this section, we will firstly discuss the observational properties of the central galaxies. We will then move to the luminosity functions, galaxy substructures, and the large-scale structure around FGs. 4.2.1. Central galaxies: formation scenarios Central galaxies in clusters are a unique class of objects. They are usually the largest and most luminous member galaxies and they lie very close to the peak of the cluster X-ray emission (e.g. Jones & Forman, 1984;Lin et al., 2004). Moreover, in the velocity space they sit near the rest frame velocity of the cluster (Quintana & Lawrie, 1982;Zabludoff et al., 1990;Oegerle & Hill, 2001). These characteristics imply that the BCGs are located at the minimum in the cluster potential well. Zarattini et al. (2019) found that there is a dependence of the velocity segregation on the magnitude gap. This result means that BCGs in FGs are located closer to the minimum of the cluster's potential well, when compared to BCGs in non-fossil systems. This difference is not found for satellite galaxies, independent on their mass.
The formation of central galaxies in FGs is thought to be the end result of the group evolution (Ponman et al., 1994;Jones et al., 2003). The main actor, in this scenario, would be dynamical friction. However, Sommer-Larsen (2006) suggested that the main difference between FGs and non-FGs has to be found in the initial velocity distribution. In particular, he found that satellite galaxies in FGs should be located on more radial orbits than in non-FGs. This would favour low angular momentum mergers, for which dynamical friction could be more effective (Lacey & Cole, 1993).
Méndez-Abreu et al. (2012) analysed the photometric properties of central galaxies in FGs. They studied the position of the central galaxies in FGs in the fundamental plane and its projections to explain the formation of these objects (see Fig. 5). Central galaxies in FGs results to have large K-band luminosities. The K s luminosity is a good proxy of the total stellar mass since the typical M/L ∼ 1 for an old stellar population (Bruzual & Charlot, 2003). Thus, it seems clear that central galaxies in FGs are amongst the most massive galaxies known. The left panel of Fig. 5 shows the correlations between the K s −band luminosities and the central velocity dispersion (Faber-Jakson relation, Faber & Jackson, 1976). Similarly, the right panel of Fig. 5 shows another projection of the fundamental plane, the K s −band luminosity vs. effective radius (r e ). In both relations BCGs in FGs are found slightly outside the correlations: this bend was found for the first time in Bernardi et al. (2011) for early-type galaxies and it can be interpreted in terms of the formation scenario of the BCGs. In particular, if major dissipationless mergers between galaxies are the main mechanisms to build up the mass of the BCGs, the final size is expected to increase, but not its central velocity dispersion; however, if minor dry mergers are the predominant mechanism, they are expected to change both the size and velocity dispersion of the BCGs. Thus, the results presented in Méndez-Abreu et al. (2012) seems to favour the first scenario for these massive galaxies. This is also supported by new results obtained from numerical simulations. Kundert et al. (2017) investigated the origin of FGs in the Illustris simulation. With respect to the stellar mass assembly of the BCGs, they found (see their fig. 6) that the accretion is similar at high-redsfhits for fossil and non-fossil systems, whereas a clear difference start to appear at z ∼ 0.3. From this point, BCGs systematically accreted more mass in FGs than in non-FGs.
However, this formation scenario is in contradiction with the one proposed by Khosroshahi et al. (2006): in fact, these authors found disky isophotes in their central regions of BCGs, in contrast with most BCGs in non-fossil systems (Rest et al., 2001). This result seems to favour a scenario in which mergers in FGs were rich in gas, thus including large M * spirals. A similar result was found also by Eigenthaler & Zeilinger (2009), since they found many shells around central galaxies in FGs. These shells are likely formed recently via major mergers of spiral galaxies (Hernquist & Spergel, 1992).
Signs of recent mergers were indeed found by Alamo-Martínez et al. (2012) while analysing the surface brightness profiles of 3 FGs. However ,this work was mainly focused on the study of globular clusters in FGs. These objects are powerful tools to study galaxy assembly, since they are old and dense enough to survive galactic interactions. Alamo-Martínez et al. (2012) results seem to point out that globular clusters in BCGs formed in a similar way in fossil and non-fossil systems. In terms of the formation scenario of the BCGs, this can be take as a confirmation that similar processes are at work in the formation of BCGs in fossil and non-fossil systems, although more statistics is needed to generalise this conclusion. A similar result, in terms of the formation scenario, is found also in Madrid (2011) and Madrid & Donzelli (2013), in which the authors compared the properties of ultra-compact dwarf galaxies (UCDs) in FGs and in the Coma cluster. UCDs are considered the bright-and-massive tail of the globular cluster distribution. Their results showed that UCDs are likely to be a common occurrence in all environments.
The presence of AGNs can also be used to study the formation scenarios of BCGs. In fact, AGNs need major mergers to form, since they use the gas provided by the merger as their fueling mechanism. On the other hand, if no other merger occurred, they are destined to end this fuel and inactivate. Hess et al. (2012) studied the sample of 34 FG candidates from Santos et al. (2007) using radio observations in order to detect the presence of AGNs in FGs. They found that 67% of these FG candidates contain a radio-loud AGN. This result seems in contrast with the old formation expected for FGs: in fact, AGN should have run out of fuel since FGs's last major merger. For this reason, Hess et al. (2012) suggested that other mechanisms, such as minor mergers, cooling flows or late time accretion should be invoked to keep the AGNs alive in FGs. However, it is worth noting that the Santos et al. (2007) sample was not pure and about half of the sample was formed by non-fossil systems (Zarattini et al., 2014).

Central Galaxies: stellar populations properties
We already mentioned that the formation scenario proposed by Ponman et al. (1994) supposed that FGs formed at high redshift, with few interactions with the large-scale structure along their lives. This would leave enough time for the M * galaxies to merge with the BCG, thus forming the ∆m 12 gap. On the other hand, Mulchaey & Zabludoff (1999) suggested the so-called failed group scenario, in which the BCG is formed as a local over density and no other bright galaxy formed within the group.
However, these two scenarios should also leave clear imprints in the stellar populations of the BCGs. In fact, in the failed group scenario the central galaxy formed via monolithic collapse, that is expected to create large radial metallicity gradients in the distribution of the stars. On the other hand, in the merging scenario the BCG suffered various major mergers, that have the power to erase those gradients, since they mix up the stars and gas during the merging process (e.g. Pipino et al., 2010). La Barbera et al. (2009) were the first in using stellar populations to investigate difference between FG's BCGs and regular elliptical galaxies. They used spectra from the SDSS DR4 for their sample of 25 BCGs in FGs and 17 field elliptical galaxies, that act as the control sample. They searched for the single-stellar-population model that best fits the spectra, thus computing mean ages, metallicities, and α−enhancement for the two populations of BCGs. The authors showed that no significant difference is found in these paramenters and concludes that BCGs in FGs did not form earlier than the other galaxies. Harrison et al. (2012) select a sample of 17 FG candidates by combining XMM observations with SDSS DR7 data. They analyse the stellar populations of the BCGs of these systems using SDSS spectra and the Starlight code (Cid Fernandes et al., 2005). Their results showed no significant differences in stellar star formation rates, age, and metallicities between FGs and their two control samples, one built by optically-selected BCGs, the other with X-ray-selected BCGs. Eigenthaler & Zeilinger (2013) studied the presence of such gradients in age and metallicity for a sample of six BGCs in FGs using longslit spectroscopy from the 4.2m William Herschel Telescope (WHT). They found that the metallicity gradient is flatter with respect to the predictions of the monolithic collapse (∼ −0.2 instead of −0.5), thus indicating the presence of mergers during the life of the BCGs. On the other hand, the age gradient is, on average, negligible.
A similar study was performed by Proctor et al. (2014) for a sample of 2 central galaxies in FGs, using longslit spectroscopy obtained with the 8m Gemini North telescope. The authors found different results for the two BCGs: SDSS J073422.21+265133.9 showed a strong metallicity gradient and a slightly positive age gradient, suggesting a relatively recent episode of stellar formation in the centre. NGC 2484, on the other hand, showed an old stellar population (∼ 10 Gyr) and a flat central metallicity, that was interpreted as the evidence of an inside-out stellar formation, at least in the final episode of stellar formation.
Trevisan & Mamon (2017) studied a large sample of 550 groups to characterise the dependence of the stellar populations of the BCGs and the second brightest galaxies with the magnitude gap. They did not find differences in the distribution of colours, star-formation rates, α−enhancement, age, metallicities and star-formation histories in systems with different ∆m 12 . Corsini et al. (2018) studied both the stellar populations and radial gradients for a sample of two BCGs in FGs using the 10.4m Gran Telescopio Canarias (GTC) telescope. They confirmed the results of La Barbera et al. (2009) and Eigenthaler & Zeilinger (2013) out to the effective radius. They found an underlying and diffuse older stellar population, with a younger one located near the centre of the galaxies. This was interpreted as the sign of the last major merger with gas, occured ∼ 5 Gyr ago. Corsini et al. (2018) also found a radial metallicity gradient in agreement with the Eigenthaler & Zeilinger (2013) one.
Finally, Raouf et al. (2019) divided a sample of groups from the Galaxy And Mass Assembly (GAMA) survey into relaxed and unrelaxed, using ∆m 12 and the luminosity offset as the relaxation indicators. They found that BCGs in unrelaxed systems are bluer, more star forming, and with non-elliptical morphologies than those in relaxed systems. They conclude that the higher rate of recent mergers expected in unrelaxed groups could be responsible for these differences. A similar result was also found in Pierini et al. (2011). These authors claimed that there are few star-forming galaxies in FGs, making them more mature then coeval and similar mass groups.
Very recently Raouf et al. (2021) studied the kinematic of gas and stars in 154 central galaxies taken the Sydney-AAO Multi-object Integral field (SAMI) galaxy survey. In particular, they divided this sample into low and high luminosity gap system, with the latter that can be assimilated as FGs. They found that there is a weak statistical difference (at approx. 1−σ level) between the magnitude gap and the gas-star kinematics misalignment. In addition, a similar difference was observed between the magnitude gap and the regularity of the stellar rotation of the BCGs. In particular, systems with high magnitude gaps are found to be more regular rotators and with a smaller fraction of gas-star misaligned kinematics.
These studies did not find relevant differences in the stellar populations of central galaxies in fossil and non-fossil systems. The imprint of monolithic collapse is not found, all the observations point towards the creation of FGs via the merging scenario, in which the gap is created via major mergers of M * galaxies. The possibility of the existence of a fossil phase in the life of a cluster is supported also by the presence of younger stellar populations in the centre of galaxy, probably due to recent major mergers with gas. In the top panel of Fig. 6 we show the relation between central velocity dispersion and central metallicity for a sample of ten FGs taken from Eigenthaler & Zeilinger (2013); Proctor et al. (2014); Corsini et al. (2018) and compared with the sample of normal and dwarf early-type galaxies of Koleva et al. (2011). In the lower panel of the same figure, the comparison is done for the central ages of the same samples. It can be seen that in both cases FGs are found in the same correlations as normal early-type galaxies. This plot again confirms that central galaxies in FGs are amongst the most massive known in the Universe.

Luminosity functions
The luminosity function (hereafter LF) is one of the most powerful tools to study the galaxy population of a group/cluster of galaxies. It is given by the number density of galaxies per luminosity interval and it is usually described parametrically with the Schechter function (Schechter, 1976). The main parameters are the characteristic magnitude (M * ) and the faint-end slope (α). The former describes the bright part of the LF, whereas the latter is related to the dwarf galaxy population. A debate is ongoing on the universality of the LF: in fact, photometric studies found that LFs in clusters are steeper than in the field (−2.0 < α < −1.8 in clusters and −1.5 < α < −1.3 in the field, see Popesso et al., 2006;Blanton et al., 2005). On the other hand, spectroscopic studies found no differences in the α parameters of the field and clusters, finding a general value of α ∼ −1.3 (Aguerri et al., 2020, and references therein) The study of LFs in FGs was mainly focused on individual FGs, due to their paucity. As a consequence, most of the first results were contradictory. Mendes de Oliveira et al. (2006) found, for the FG called RX J1552.2+2013, M * = −21.18 ± 0.57 and α = −0.77 ± 0.37 using spectroscopically-confirmed members, or M * = −21.27 ± 0.62 and α = −0.64 ± 0.30 for photometrically-selected galaxies in the r−band. In the same year, Khosroshahi et al. (2006) computed M * = −20.40 ± 0.22 and α = −1.23 ± 0.28 for RX J1416.4+2315, a fossil system with a mass similar to RX J1552.2+2013. Also Trentham et al. (2006) presented an LF for a single FG (NGC 1407), finding α = −1.35. The difference, especially in the faint-end slope, is important. In fact, α = −1 indicates a flat LF, in which the number of dwarf galaxies is not changing with magnitude. On the other hand, in a steeper function like that of Khosroshahi et al. (2006), the number of dwarf galaxies is rapidely growing and, in a flatter one like that of Mendes de Oliveira et al. (2006) it is decreasing. Zibetti et al. (2009) also studied the photometric LF of a sample of five FGs. They found a faint-end slope in agreement with the one of regular clusters presented in Popesso et al. (2006). However, it is worth noting that Popesso et al. (2006) found an upturn at fainter magnitudes, so that their LF can be fitted with a double Schechter function. This behaviour is not found on the already-cited LF of FGs, because none of these are deep enough. However, Lieder et al. (2013) computed a very deep LF for NGC 6482 using spectroscopic data from the Subaru/Suprime-Cam, finding α = −1.32 ± 0.05. They did not fit a double Schechter function, however a change in the faint-end slope is present also in their fig. 12. Aguerri et al. (2011) presented the LF of RX J105453.3+552102, a massive FG at z = 0.5. They found M * = −20.86 ± 0.26 and α = −0.54 ± 0.18, thus confirming a flatter trend for the dwarf galaxy population. Adami et al. (2012) also computed the LF for two FG, finding that their faint-end slope is relatively flat, but without giving numbers. Finally, Aguerri et al. (2018) studied the spectroscopic LF of RXJ075243.6+455653, finding α = −1.08 ± 0.33.
The first systematic study of the dependence of the LF on the magnitude gap was presented in Zarattini et al. (2015). The authors selected ∼ 100 clusters and groups spanning a wide ∆m 12 range and dividing their analysis in four bins of ∆m 12 . Their study was based on a hybrid method for computing the LF, in which the bright part was treated as a quasi-spectroscopic LF, whereas in the faint end photometric data were dominant. The authors computed a classical LF and one in which the magnitudes of each systems are referred to the magnitude of the central galaxy (e.g. M r − M r , BCG, called relative LFs). The latter permits to compare directly the differences due to the magnitude gap and the authors found that this technique offers the best results for highlighting the differences between their four subsamples. These relative LFs are shown in the left panel of Fig. 7. Zarattini et al. (2015) found that both M * and α changes with the magnitude gap. In particular, systems with ∆m 12 < 0.5 have the brightest M * and the steepest α slope, whereas systems with ∆m 12 > 1.5 have the faintest M * and the flattest α. The differences are larger than 3σ between the two most-extreme cases and the trend with ∆m 12 is clearly visible in the right panel, where the value of the relative M * and α are shown for the four subsamples.
The discussion is usually focused on the faint end of the LF, since differences in the bright part between fossils and non-fossils can be easily explained with the same mechanisms that are responsible for the creation of the magnitude gap. Moreover, the presence of the gap itself as a selection criteria implies differences in the bright part of the LFs.
On the other hand, the debate on the differences in the faint-end slope is more complex. In fact, dwarf galaxies can not be merged into the BCG in a reasonable time, since the merging time scale is inversely proportional to the mass of the satellite. A possible explanation is that dwarf galaxies are located in radial orbits passing close to the centre of the cluster. This will result in their disruption, accounting for a part of the missing population. This explanation was also invoked as a possible reason for the high merging rate  Zarattini et al. (2015). Empty black squares are systems with ∆m 12 < 0.5, filled red circles are systems with 0.5 < ∆m 12 < 1.0, empty violet circles are systems with 1.0 < ∆m 12 < 1.5, and filled green circles are systems with ∆m 12 > 1.5. Right panel: Uncertainty contours for the Schechter fits of LFs in the left panel. Contours represent 68%, 95%, and 99% c.l. and the colour and symbol codes are the same as in left panel. Original image can be found in Zarattini et al. (2015), in particular left panel correspond to their figure 7, right panel to figure 8. of FGs. It can also justify the formation of the magnitude gap, since Lacey & Cole (1993) showed that the merging timescale for satellite on radial orbits is shorter than for tangential ones. Another possibility is that FGs could lack dwarf galaxies for differences in their accretion time. In fact, Aguerri et al. (2018) found that the large-scale environment of FGS03 is very rich, so the flat LF found could be explained if the dwarf population are still trapped in nearby groups, awaiting to be merged with the FG. In this context, FGs would be systems in early stages of their mass assembly. We will discuss in more details the large-scale structure of FGs in Sect. 4.2.5.
Differences in the accretion history of dwarf galaxies were indeed found also in Kanagusuku et al. (2016) while studying the bright and dwarf galaxy populations in fossil and non-fossil clusters in the Millennium simulation. They found that FGs had a denser dwarf population at early time (z > 0.7), then the trend reverses (0.5 < z < 0.3), and finally it become similar at z = 0.
To finally solve the issue of the observed lack of dwarf galaxies in FGs, deep and extended spectroscopy would be needed. We will discuss in Sect. 6.2 the expected impact on this topic of the next-generation spectroscopic surveys.

Galaxy substructures
If the old and relaxed model is correct, one should expect to find a smaller amount of galaxy substructures in FGs than in non-FGs. The only study on this topic was done in Zarattini et al. (2016). They analysed the sample of 34 FG candidates of the FOGO project to compute the fraction of FGs with signs of substructures using a variety of methods. In particular, the entire sample was studied with a two-dimensional approach, able to detect substructures in the projected space of the cluster. Moreover, for a subsample of candidates for which an extended spectroscopic follow up was available, they also applied a series of one-and three-dimensional tests (e.g. the Dressler-Schectman test). Zarattini et al. (2016) results depend critically on the adopted tests, but the comparison with a control sample shows that galaxy susbstructures are present in a similiar fraction in fossil and non-fossil systems. This presence of substructures in FGs is hardly compatible with an old formation, followed by a passive evolution, with no major interactions with the surrounding large-scale structure. Indeed, the small number of genuine FGs in the sample prevent to reach a definitive conclusion.

Large scale environment
Differences in the large-scale environment and in the way in which it interacts with FGs were invoked as a possible cause of the different evoulution of FGs and non-FGs (D'Onghia et al., 2005a;Ponman et al., 1994;Díaz-Giménez et al., 2011). Observational results are scarce on this topic, since only few individual FGs were studied so far. Adami et al. (2007) studied the large-scale structure around the FG RXJ1119.7+2126 using spectroscoic data. They conclude that this FG is located at the centre of a low galaxy density bubble. Pierini et al. (2011) found controversial results in their analysis of two FGs. One of the two is found in an isolated environment, whereas the second one is located in a dense environment, with 27 other groups or clusters in the surroundings.
Also Adami et al. (2012) studied the environments of other three FGs using photometric and spectroscopic data. They found that one system (1RXS J235814.4+150524) is in a poor environment, though its galaxy density map shows a clear signature of the surrounding cosmic web. The second FG (RX J1119.7+2126) is very isolated, whereas the third one (NGC 6034) is embedded is a very rich environment.
Finally, Díaz-Giménez et al. (2011) analysed the large-scale structure in FGs and non-FGs from both the theoretical and observational points of view. We already mentiond their theoretical results in Sect. 3, here we focus on their observational tests. In fact, they used four FGs selected from Voevodkin et al. (2010) and coming from the 400d cluster survey (Burenin et al., 2007) and a control sample of non-FGs from the same survey. Their observational results confirmed the peak found in numerical simulations in the local density profile of galaxies around groups as a function of the normalised group-centric distance. This peak, located at about 2.5r/r vir at z = 0, is more prominent in FGs. However, the difference is clearer in numerical simulations than in observations, probably again due to the small sample of FGs available.
More extended studies on larger samples are thus generally needed to confirm if FGs are characterised by a special large-scale environment.

Past and future of fossil systems evolution
If FGs are the end product of groups/clusters evolution, a question should naturally arise: are their progenitors regular groups/clusters or do they belong to some particular class? A possible answer, that we will discuss in the first part of this section, is that the progenitors could be found in compact groups. In fact, in these systems various bright galaxies are cooped up in a small area, making them ideal candidates for fast and efficient mergers.
However, von Benda-Beckmann et al. (2008) suggested that the fossil status may be only a transitional phase in the life of a regular cluster. If this is the case, there is no need to find a special category of progenitors, since the acquisition of the fossil status could happen to any group/cluster in the period between the last major merger and the subsequent arrival of another bright galaxy from the cosmic web. We will discuss this topic from an observational point of view in the second part of this section.

Compact and loose groups as progenitors of fossil systems
The discussion on the progenitors of FGs started even before the actual discovery of these systems. In fact, Ponman & Bertram (1993) suggested that compact groups were the result of orbital decay in larger systems and that they should culminate in a final merger, in which all the bright galaxies would merge at the centre of the system. They also added that a new category of "fossil groups" were awaiting discovery in the ROSAT all-sky survey. A year after this claim, they announced the discovery of the first FG (Ponman et al., 1994).
Since then, many studies looked for FG's progenitors, using a variety of techniques. Miles et al. (2004) studied a sample of 25 clusters in the X-ray, finding that some of them are dimmed in luminosity. Their interpretation of the result was that, according to a specific toy simulation, groups with dimmed X-ray luminosity have lower velocity dispersion. This would lead to the formation of FGs, since low-velocity encounters between massive galaxies are the most efficient in terms of merging time scale. This result was confirmed also by the analysis of a compact group at z = 0.22 studied in Mendes de Oliveira & Carrasco (2007). In fact, these authors showed that this system has many characteristic in common with FGs and, in particular, the merging of the four brightest members would lead to the formation of a BCG of M r ∼ −23, a typical value of central galaxies in FGs. A similar result was also found in the study of galaxy pairs (Grützbauch et al., 2009). In this case, the author claims that E+S pairs could be the last step in the formation of FGs, thus a sort of transition between compact groups and FGs. Also Pierini et al. (2011) found evidences that compact groups are favoured as progenitors over the early assembly. On the other hand, Yoshioka et al. (2004) found a M/L ratio for FGs that is too high if compared with compact groups, thus claiming that these systems are not the ideal progenitors.
A novel technique to search for FG's progenitors is the use of strong gravitational lensing in galaxy groups. Johnson et al. (2018b) investigated the fraction of FGs in lensed and non-lensed galaxy groups, finding that the fraction of FGs is larger in lensed systems (13% versus 3%). They also identified 12 possible FG progenitors, that were later investigated in details in Johnson et al. (2018a) using Chandra and the Hubble Space Telescope. Their results showed that the X-ray temperatures of the candidate progenitors are higher than those of the control sample. They also find hints of differences in the LFs of FGs and non-FGs, but these differences are erased when BCGs are removed. Finally, Schirmer et al. (2010) studied the strong-lensed FG J0454-0309, that is found behind a well-studied non-fossil poor cluster. Their analysis support a scenario in which the fossil system is falling into the poor cluster and where the central galaxy of the FG will become the brightest galaxy of the new system.
Another approach was proposed by Tovmassian (2010): they compared the K−band absolute magnitudes of BCGs in regular clusters and FGs, finding that the latter are systematically fainter. The author concluded that FG progenitors are likely poor groups. Moreover, it is interesting to note that Tovmassian et al. (2006) studied "the properties of Hickson's compact groups and of the Loose Groups within which they are Embedded". In this work, no link with FGs is suggested, but the result could be seen under a different light after the Tovmassian (2010) study, confirming the idea that poor compact groups could be the ideal progenitors of FGs. However, this result apparently collides with what was presented in Sect. 4.2, where we cited various studies claiming that BCGs in FGs are amongst the most-massive galaxies in the Universe. On the other hand, Farhang et al. (2017) analysed the mass assembly histories fo compact and fossil systems in the Millennium simulation and associated semi-analytical models. They found that only 30% of FGs could originated from compact groups. They conclude that most of the fossil and compact groups follow different evolutionary paths.
Finally, Buote & Barth (2018) suggested that compact elliptical galaxies (CEGs) surrounded by an X-ray halo should be considered as FGs. In fact, detailed X-ray observations of two of these systems found that mass and entropy profiles and concentration are compatible with other FGs studied in the literature. In this case, the progenitors are expected to be the so-called "red-nuggets", compact galaxies found a z ∼ 2 Ferré- Mateu et al. (2017). This case is indeed peculiar: the authors did not study FGs in order to find their progenitors, but better suggested that a new type of galaxy should be considered in the fossil category.

Transitional fossil phase
We already mentioned in Sect. 3 that von Benda-Beckmann et al. (2008) suggested that FGs are only a transitional phase in the life of a regular group/cluster. They claimed that this phase would happen just after a major merger and before other bright galaxies are accreted to the group. An example of such a process is found in Irwin et al. (2015): the Cheshire Cat galaxy group is formed by two smaller groups, dominated by one bright galaxy each. These groups are experiencing a line-of-sight merger that will end up in approximately one Gyr with the merging of the two structures. The authors suggested that the resulting structure will be a massive fossil group, dominated by a large M r = −24 galaxy.
A similar case is the one presented in Aguerri et al. (2018): RX J075243.6+455653 was found to actually accomplish the fossil definition of ∆m 12 > 2 within half the virial radius, however another galaxy almost as bright as the BCG is found just outside that radius. Depending on its orbit, RX J075243.6+455653 became fossil in the very last part of its life or, in the opposite case, it will become non-fossil in the near future.
The existence of a fossil phase may thus explain some of the controversial results presented along this review. It is possible that the observational definition based on the magnitude gap alone is not sufficient to clearly separate the population of real FGs to that of non-FGs dominated by a massive central galaxy. This would confirm the results of Raouf et al. (2014), since the authors suggested that other observational quantities (like the luminosity of the BCG and its separation from the luminosity centroid of the group) should be used to create a sample dominated by purely old FGs. We thus suggest to start using this new definition in the search for FGs as a way for creating a sample of old systems. However, using these additional observational constraints could dramatically reduce the number of identified systems. In Sect. 6.1 we will give a list of the most-secure FGs up to date: only one out of 18 FGs for which the absolute magnitude of the BCG is available will survive the application of the Raouf et al. (2014) criterium (∆m 12 > 2.0 and M r,BCG > −22.5).

Discussion and conclusions
In this section we propose a sample of genuine FGs that can be the starting point for new follow ups of these objects. Then we discuss what we presented along the review and draw our general conclusions.

Sample of genuine fossil groups
In Table 1 we present a list of confirmed FGs in the literature. The goal is to offer to the reader a sample as pure as possible for future follow ups. The list is probably not complete, but we did our best to select FGs applying a rigorous criterium on the ∆m 12 parameter. In particular, we consider as fossils those systems with ∆m 12 ≥ 2.0 within half the (projected) virial radius. Moreover, we exclude FGs for which membership was done using a fixed cut in ∆z, except when no ambiguous galaxy was found within half the (projected) virial radius. The magnitude gap between the first and fourth brightest galaxies (∆m 14 ) is also given, when available, but it was not used for the selection, since it was computed only in the most recent studies. The absolute r−band magnitude of the BCG is included, when available, to simplify the application of the Raouf et al. (2014) criterium. Moreover, we also list the mass of the system, when available. However, we note that the mass is computed in a very inhomogeneous way (different methods and radii), our goal is to offer an at-a-glance reference to the reader. Finally, the redshift is given for all FGs listed in the table.
We note that, if contradictory information are available, we always choose to apply the Jones et al. (2003) criteria in the most severe way. For example, in Proctor et al. (2011) the sample of Miller et al. (2012) was studied in more details, computing r 200 in two different ways: one obtained from weak lensing analysis and the other from X-ray data. The latter is found to be ∼ 50% larger than the former. As a consequence, the number of FGs found using the smallest radius is 10, a number that reduces to 3 if the It is worth noting that some famous FGs are excluded from the list. As an example, we discuss the prototype of this category, NGC 1132 (Mulchaey & Zabludoff, 1999), for which we were not able to find ∆m 12 . There are indeed information on the fainter galaxies (e.g. Kim et al., 2018), but not a clear computation of the magnitude gap. However, Kim et al. (2018), claimed that the second brightest galaxy is NGC 1126, a spiral galaxy located at 8.4 arcminutes or 230 kpc in projection. The virial radius of this group is estimated to be r 200 ∼ 800 kpc, thus this galaxy should be inside 0.5 r 200 . The difference in the velocity space is ∆v = 438 km s −1 , as obtained using the Nasa Extragalactic Database (NED), so the two galaxies can be part of the same group, although a precise dynamical study should be done to confirm the membership. In Kim et al. (2018) NGC 1126 is described as "seven times faniter in B". We check in the SDSS DR16 the magnitudes of both NGC 1132 and NGC 1126: the former has m r = 12.20, the latter m r = 14.01, leading to a ∆m 12 = 1.81, rejecting it as a genuine FG. However, as we already mentioned, an accurate study of the membership of NGC 1126 to the group of NGC 1132 should be done and errors in SDSS magnitudes can not be excluded (NGC 1126 is flagged with "unreliable photometry", as many other bright galaxies, mainly due to an over estimation of the sky around bright objects). Concluding, with this example we aim at demonstrating that also the definition of the most commonly-accepted FGs may be not rigorous, or may need deeper studies to include them in a sample of genuine FGs.

Conclusions and future prospects
Along this review we analysed the most-studied topics on FGs. The aim of these studies was to test the so-called merging scenario, which predicts that fossil systems formed earlier than non-fossils, having enough time to merge all the bright galaxies with the BCG and remaining somewhat isolated from the cosmic web (e.g. they did not receive other bright galaxies from the merging with other groups/clusters). However, the general framework that can be obtained from this review is that FGs probably formed and evolved in a similar way as non-FGs. In particular, we show that early differences reported in global properties such as the scaling relations and M/L ratios of the halos of fossil systems can be reconciled when homogeneous datasets are used. Probably, an analogous result would be obtained for the differences observed in the mass and the entropy profiles in some individual systems. A more homogeneus and large sample is required to be analysed in this case. Moreover, no differences are found in the fraction of galaxy substructures identified in FGs and non-FGs. This again indicates that the halos of FGs are not significantly older than those from non-FGs.
The central galaxies in fossil groups show similar stellar ages and metallicities than BCGs in the center of non-fossil systems. In addition, the location of these galaxies in the fundamental plane and its projections indicate a formation process driven by dissipanionless mergers in a similar way as other bright early type galaxies.
The similarities found in the formation of fossil and non-fossil systems seem to indicate that the large magnitude gap could just be a transient phase in the evolution of groups and clusters, as reported by different numerical simulations. This magnitude gap would be more connected with recent major mergers rather than with an old formation.
If this is the case, one should find an explanation for those differences that can not be reconciled with inhomogeneities in the data. We already mentioned that Sommer-Larsen (2006) proposed that more radial orbits for galaxies in fossils could be responsible for the formation of the gap. This idea is supported also by Lacey & Cole (1993): the merger timescale with the central halo is shorter for M * galaxies on radial orbits than for galaxies on tangential orbits (see their eq. 4.2). From an observational point of view, hints of this difference are found in Zarattini et al. (2021, submitted to A&A). The authors studied the orbital structure of a sample of ∼ 100 groups and clusters, dividing them in four bins of ∆m 12 . Their larger magnitude gap bin (∆m 12 > 1.5) shows the presence of radial orbits in the external regions (0.8 − 1 r 200 ), that is not found in the other three bins, all with ∆m 12 < 1.5. However, the results should be confirmed with a larger sample of genuine FGs, as we already mentioned along this review for a significant part of the discussed topics.
The other main topic that remains open is the difference found in the faint-end slope of FGs LFs. This is difficult to explain within the current models of formation and evolution of clusters. In fact, most of the studies points toward a sort of global value for the faint-end slope in clusters and groups. It is worth noting, however, that the majority of the studies of LFs in FGs used photometric data and that, in the literature, significant differences were found even in regular clusters when only photometric data were used. Thus, the next step in this discussion awaits the use of large spectroscopic datasets, that will become available with the next generation of astronomical instruments (e.g. WEAVE, 4MOST, DESI). However, we can tentative say that the presence of radial orbits could give an answer also to the problem of the faint-end slope of the LF: in fact, if massive galaxies on radial orbits have a shorter timescale to be merged within the central galaxy, dwarf galaxies could be more easily disrupted on such orbits, if they pass near the BCG (Smith et al., 2015). Another possible explanation for the differences in the galaxy populations between FGs and non-FGs could be found in the surrounding environment, since hints of different large-scale structures are found in some individual studies. In particular, we can not exclude that FGs are still in the process of accreting dwarf galaxies, but the general picture remains to be clarified.
The creation of a large and strict sample of genuine FGs and a homogeneous follow up will be the key of the characterisation of FGs in the near future. For this reason we gave in Sect. 6.1 a table with the most-secure FGs to date. The computation of the ∆m 12 is not a real issue with surveys like SDSS, DES, or Pan-STARRS1, already available for 3/4 of the sky. The arrival of new facilities will be useful for the confirmation of the FGs candidates found with these photometric surveys. New X-ray data will be available with the new all-sky surveys like eROSITA (we refer the reader to the companion review of Eckert et al. (2021) for a detailed description on the impact of eROSITA and other X-ray surveys on the study of galaxy groups). In addition, the spectroscopic follow up will be possible with extended spectroscopic surveys like WEAVE, 4MOST, and DESI or with precise photometric redshifts surveys like J-PAS. The firm identification of at least 50/100 FGs will be the main scientific goals in this field for the next decade. This will be easily achieved in the near future, since eROSITA is exepcted to find ∼ 10 5 groups/clusters. For comparison, the REFLEX cluster catalogue has ∼ 1500 groups/clusters. Most of these new clusters will have dedicated spectroscopic follow ups with the next generation multi-object spectrographs.