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Classification of non-solvable groups whose power graph is a cograph

Cameron, Manna and Mehatari investigated the question of which finite groups admit a power graph that is a cograph, also called power-cograph groups (Journal of Algebra 591 (2022)). They give a classification for nilpotent groups and partial results for general groups. However, the authors point out number theoretic obstacles towards a classification. These arise when the groups are assumed to be isomorphic to PSL 2 (q) or Sz(q) and are likely to be hard. In this paper, we prove that these number theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 (q) or Sz(q). For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg-Kegel graph is connected. Moreover, we reduce the case where the Gruenberg-Kegel graph is disconnected to the classification of p-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.


Introduction
A classic reoccurring approach to investigating the structure of groups is by associating the groups with graphs and then studying their graph theoretical properties. Prominent examples include Cayley graphs, capturing the structure of groups relative to the action of a specified generating set, or orbital graphs, capturing primitivity properties of permutation groups.
A central example in the context of the present work is the Gruenberg-Kegel graph, also called the prime graph of a group. Rather than describing actions of groups, the prime graph π(G) of a group G is built from abstract structural aspects of the group: the vertices are the primes dividing the order of the group and two primes are joined by an edge if and only if their product is realized as the order of some group element. The prime graph was first investigated by Gruenberg and Kegel (see [15], some results were later published by Williams in [38] based on an unpublished manuscript by Gruenberg and Kegel) in the context of augmentation ideals of integral group rings. The structure of the prime graph has deep connections with the structure of the group itself. For instance, if π(G) is disconnected, then the structure of G is quite restricted (cf. [24,Theorem 14]) and if G is solvable, then π(G) does not contain an induced coclique of size 3 [23].
A recent, active branch of research studies graphs whose vertex sets are themselves groups. A basic but important example in this scope is the power graph of a group, which was first studied in the context of building graphs from semigroups [20]. The vertices are given by an underlying semigroup and arrows are drawn from elements to all their powers. Power graphs have been frequently studied throughout the literature, see for example [1,2].
A comprehensive overview of various graphs defined on groups including many key results and open problems can be found in [7]. There, Cameron points out that many commonly investigated graphs defined on groups fit into a hierarchy of subgraphs. The hierarchy includes the (directed) power graph and the enhanced power graph, the commuting graph and the deep commuting graph and the generating graph. An extension of this hierarchy into a second dimension has been studied in [3]. A particular focus lies on the interactions between different graphs. For instance, it is known that all graphs from the hierarchy uniquely determine the prime graph [7,Theorem 2.9].
The work of Cameron et al. has spawned a number of intriguing problems relating the structure theory of graphs and groups as well as problems of classifying groups in terms of restrictions on their graphs, often resulting in new characterizations of interesting known group classes. There has been ongoing research to determine the groups for which two of the graphs in the hierarchy coincide, and various conditions and results have been proven and summarized in [1,7]. While investigating groups that are unrecognizable by their prime graph, Cameron and Maslova recently gave a classification of non-solvable EPPO-groups [9]. EPPO-groups are groups in which every element has prime power order. The study of such groups was initiated by Higman in [16], and later extended and generalized by numerous authors, see for example [6,31], including the famous work of Suzuki [34,35] on doubly transitive groups.
In the present work we are concerned with generalizing the classification of EPPOgroups by extending it to a classification of groups whose power graph is a cograph (in the following called power-cograph groups). The class of power-cograph groups indeed contains all EPPO-groups by an argument of Cameron [7]. A cograph is a graph that does not contain a path on four vertices as an induced subgraph. Cographs are frequently studied in structural graph theory and have been rediscovered numerous times, see for example [19,30,33,11]. An important theme in structural graph theory is the study of hereditary graph classes which can be defined via forbidden induced minors [4]. After cluster graphs (defined by forbidding induced paths on three vertices), cographs form a basic and natural class of graphs to consider, as they are defined by a single forbidden induced subgraph.
The problem of classifying power-cograph groups goes back to [24], where the authors investigate general forbidden substructures in power graphs. They also point out that classifying such groups when isomorphic to PSL 2 (q) is equivalent to a number theoretic problem that is likely to be hard. Further investigations in [8] show that the classification of simple power-cograph groups completely reduces to similar number theoretic problems for groups isomorphic to PSL 2 (q) or Sz(q) (see [8,Theorem 1.3] and the discussion after). Nilpotent power-cograph groups are fully classified and partial results are available for the solvable case in [8]. To this end, the best we can expect is a classification of nonsolvable power-cograph groups, relative to the evident number theoretic obstacles, which is what we provide in this paper. Given a solution to these problems, our results reveal a complete classification of non-solvable power-cograph groups. In particular, for the non-solvable case we prove the following theorem. Theorem 1.1 (Non-solvable power-cograph groups). Let G be a finite group that is not solvable. The power graph of G is a cograph if and only if one of the following holds: 1. G is a simple power-cograph group isomorphic to one of PSL 2 (q), Sz(q) or PSL 3 (4).
The admissible values of q are precisely characterized in [8], see Theorem 3.6 below.
3. T := G/ Soc(G) ∼ = PSL 2 (2 n ) with n ≥ 2 and T is a power-cograph group. Furthermore, Soc(G) = O 2 (G) and each minimal normal subgroup of G is isomorphic to the natural module over the group ring F 2 n [SL 2 (2 n )] as a T -module. 4. T := G/ Soc(G) ∼ = Sz(2 2e+1 ) with e ≥ 2 and T is a power-cograph group. Furthermore, Soc(G) = O 2 (G) and each minimal normal subgroup of G is isomorphic to the natural module over the group ring F 2 2e+1 [Sz(2 2e+1 )] as a T -module.
Moreover, our techniques also yield new insights for the solvable case. We obtain a complete classification of solvable power-cograph groups with connected prime graph. In the disconnected case one problem remains, and it already arises in the case of solvable EPPO-groups. In general, much is known about the structure of such groups since they are always Frobenius or 2-Frobenius groups. However, to date there is no complete classification of solvable Frobenius groups available. In particular, the question of which p-groups admit fixed-point-free automorphisms is still open. However, this is the only remaining obstacle and we precisely describe the structure of solvable power-cograph groups in terms of the following theorem. Theorem 1.2 (Solvable power-cograph groups). Let G be a finite solvable group. The power graph of G is a cograph if and only if one of the following holds: 1. G has prime power order.
2. G = (C p ⋊ C q n ) × C q , and if n > 0 then C q n acts fixed-point-freely on the C p -part of Fit(G).
3. G = C pq ⋊C r n , and if n > 0 then C r n acts fixed-point-freely on C pq . Thus r n divides q − 1 and p − 1.
4. G = C rs ⋊ C pq , where pq divides r − 1 and s − 1 and C pq acts fixed-point-freely on C rs . 5. F := Fit(G) is an r-group for some prime r and H := G/F is in one of three cases: (a) H is a generalized quaternion 2-group acting without fixed points on F .
(b) F admits a group of fixed-point-free automorphisms P ∼ = C p k for a prime p = r and H is a subgroup of P ⋊ C r m . The action of C r m on P is fixed-point-free if m > 0.
(c) F admits a group of fixed-point-free automorphisms C ∼ = C pq for primes p = q with p, q = r. Here, H is a subgroup of C ⋊ C r m . The action of C r m on C is fixed-point-free if m > 0.
Note that the groups in Case 5a have been fully classified, see [26]: F is necessarily abelian and F admits a fixed-point-free action by a generalized quaternion group of order 2 n if and only if 2 n divides p k − 1 for every cyclic direct factor of F of order p k [26, Proposition 6.24]. More generally, [26] contains a classification of all groups in Case 5 when F is assumed to be abelian. In the non-abelian case there is an extensive body of research concerned with bounding the Fitting-length of G as a function of the order of a fixed-point-free automorphism of prime order [17,37,36], but no general classification is available.
Techniques Regarding our techniques, we first exactly characterize power-cograph groups in terms of properties of centralizers of their group elements. This relates back to the classification of CN-groups (groups with nilpotent element-centralizers, see [12]) and CIT-groups (groups in which involution centralizers are 2-groups, see [34,35]). The latter includes the discovery of the famous Suzuki-groups. In these works, certain restrictions on element centralizers are shown to have strong influence on the overall structure of the ambient group. In a similar fashion, the centralizer restrictions we obtain allow for nonnilpotent centralizers but are otherwise severely limiting. We are able to employ results from [25] and [32] (building on work of Suzuki in [34,35]) in dealing with extensions of 2-groups by non-abelian simple groups. We point out that same results are also employed in the classification of non-solvable EPPO-groups in [9].
Outline of the paper We give some preliminary information about notation and the relevant theory of finite groups in Section 2. Then we discuss power-cograph groups in Section 3 and we go into detail about centralizers in Section 4. For the classification, we consider power-cograph groups in terms of the solvability of the group. We discuss solvable groups in Section 5 and non-solvable ones in Section 6. The proofs of the main theorems can be found in Section 7.

Preliminaries
In the following we collect some necessary group theoretic basics. Readers familiar with finite group theory might want to skip this section. As a general reference for basics on group theory, we refer to [14] and [27].
Notation Given a set of primes π, we say a π-element in a group G is an element whose order is only divisible by primes in π. Similarly a π-group is a group that entirely consists of π-elements. We say that an automorphism of a group H is fixed-point-free if it only fixes the identity element in H. Given a normal subgroup N of a group G, then G/N acts fixed-point-freely on N if the induced action by conjugation is fixed-point-free for every non-trivial element in G/N. General extensions of H by N, i.e., groups with normal subgroup isomorphic to N and corresponding quotient isomorphic to H, are denoted by N.H. We use U G to denote that U is isomorphic to a subgroup of G.
Group structure The following theorems fundamental to group structure theory are invoked throughout this paper.
Theorem 2.1 (Schur-Zassenhaus, see [14]). Let G be a finite group admitting a normal subgroup N G, such that |N| and |G/N| are coprime. Then N has a complement in G, that is, G is isomorphic to the semidirect product N ⋊ G/N. [5]). If a finite group G is of order p a q b with primes p and q, then G is solvable.
As a consequence, the order of every finite non-abelian simple group is divisible by at least three distinct primes.
Fitting-subgroup and socle Recall that the Fitting-subgroup Fit(G) of a group G is the (unique) largest normal nilpotent subgroup in G. Furthermore, let Fit 2 (G) denote the subgroup of G that satisfies Fit 2 (G)/ Fit(G) = Fit(G/ Fit(G)). Given a prime p, we denote the (unique) largest normal p-subgroup of a group G by O p (G) (also called the p-radical ). In case that G is a finite group, it holds that is a direct product over all p-radicals in G. If G is furthermore solvable, then Fit(G) is self-centralizing, i.e., C G (Fit(G)) = Z(Fit(G)), and then G/ Fit(G) is isomorphic to a subgroup of Aut(Fit(G)).
The socle of G, denoted Soc(G), is the subgroup generated by all minimal normal subgroups of G. The socle of a finite group is a direct product of simple groups and if Fit(G) is trivial, then all direct factors of Soc(G) are non-abelian simple.

Frobenius groups
In this case, we say that G is a Frobenius group, with Frobenius complement H and Frobenius kernel K.
Let G be a finite Frobenius group with kernel K and complement H. Then the following hold (for proofs see for example [27,36,39,40]): 1. K is nilpotent and in fact K = Fit(G).
2. For all k ∈ K, we have C H (k) = 1.
3. If H has even order, then K is abelian.
4. If G is solvable, then the Sylow p-subgroups of H are cyclic if p > 2 and cyclic or generalized quaternion if p = 2.
5. If a Frobenius complement H is not solvable, then it has a normal subgroup of index 1 or 2 that is the direct product of SL(2, 5) and a metacyclic group of order coprime to 30.
A group G is said to be a 2-Frobenius group if there exist two normal subgroups F and L of G such that G/F is a Frobenius group with kernel L/F and L is a Frobenius group with kernel F . Finally, let us point out the following well known fact about 2-Frobenius groups. Outer automorphisms of PSL 2 (q) and Sz(q) We just give a brief summary here, for more details see [39]. The outer automorphisms of the classical groups generally come in three different flavors: diagonal, field and graph automorphisms.
For PSL 2 (q), the diagonal automorphism corresponds the the action of PGL 2 (q) on PSL 2 (q). It is induced by conjugation with In particular, diagonal automorphisms only exist for odd q. In case of PSL 2 (q), the graph automorphisms appear as inner automorphisms. The field automorphisms are induced by automorphisms of the underlying field F q of order q = p f , whose automorphism group is cyclic of order f , generated by the Frobenius automorphism x → x p . The induced group of field automorphisms acts on PSL 2 (q) by mapping each matrix entry to its p-th power (for any matrix representing the respective element of PSL 2 (q)).
In the case of Sz(q), the simple Suzuki groups only have field automorphisms as outer automorphisms, acting on the natural representation of Sz(q) in GL 4 (q).
The Gruenberg-Kegel graph (or prime graph) The Gruenberg-Kegel graph or the prime graph of G, denoted π(G), has vertex set the prime divisors of |G|, and two primes p and q are adjacent if and only if there exists an element of order pq in G. We refer to the following result as Lucido's Three Prime Lemma: . Let G be a finite solvable group. If p, q, r are three different primes dividing |G|, then G contains an element whose order is the product of two of these primes.
In [38], Williams classified the number of connected components of the prime graph for the simple groups of Lie type over fields of odd characteristic, alternating groups, and the 26 sporadic simple groups. The case of simple groups of Lie type in even characteristic has been treated in [21,18]. The following result for prime graphs of solvable groups is due to Gruenberg and Kegel (published by Williams in [38] and slightly modified in [24]): Theorem 3.4]). The prime graph of G is disconnected only if G satisfies one of the following: 3. |G| is even and G is an extension of a nilpotent π 1 -group by a simple group by a π 1 -group. Here π 1 denotes the component of 2 in π(G).

Power graphs and power-cograph groups
Throughout this work we only consider finite groups. The following section collects basic definitions and some known results on power-cograph groups.
Definition 3.1. The power graph Pow(G) of a group G is the undirected graph defined over the vertex set G with an edge joining g, h ∈ G if and only if g n = h or h n = g holds for some n ∈ N.

Definition 3.2.
A graph is a cograph if it does not contain P 4 (i.e., a path with four vertices) as an induced subgraph. We say a group G is a power-cograph group if Pow(G) is a cograph.
The power graph of a subgroup H ≤ G is an induced subgraph of Pow(G). In particular, the class of power-cograph groups is closed under taking subgroups. 1. Let g, h ∈ G such that g = h holds, then g and h form twins in Pow(G), i.e., they are joined by an edge and have identical sets of neighbors.
2. If G is a cyclic p-group, then Pow(G) is a clique.
In special cases the power-cograph property also propagates to quotients.
Lemma 3.4. Let G be a finite group, g 1 , . . . , g 4 ∈ G and N G. Assume that (g 1 N, . . . , g 4 N) induces a P 4 in Pow(G/N) and that (w.l.o.g.) g 2 N is a power of both Proof. We assume that gcd(|N|, |C G/N (g 2 N)|) = 1 holds. By the Schur-Zassenhaus Theorem, G contains a subgroup that is a split extension N ⋊ C G/N (g 2 N). In particular, there exists a subgroup of G that is isomorphic to C G/N (g 2 N) and the latter contains all the g i N.
Finite nilpotent power-cograph groups are characterized in [24] as follows: Theorem 12]). Let G be a finite nilpotent group. Then P ow(G) is a cograph if and only if either |G| is a prime power, or G is cyclic of order pq for distinct primes p and q.
Recent work of Cameron, Manna and Mehatari [8] investigates several graph classes in an attempt to determine groups whose power graph is a cograph. Their main result on the class of finite simple groups states the following: Theorem 3.6 ([8, Theorem 3.2]). Let G be a non-abelian finite simple group. Then G is a power-cograph if and only if one of the following holds: where q is an odd prime power with q ≥ 5, and each of (q − 1)/2 and (q + 1)/2 is either a prime power or the product of two distinct primes; where q is a power of 2 with q ≥ 4, and each of q − 1 and q + 1 is either a prime power or the product of two distinct primes; 3. G = Sz(q), where q = 2 2e+1 for e ≥ 2, and each of q −1, q + √ 2q + 1 and q − √ 2q + 1 is either a prime power or the product of two distinct primes; The discussion that follows this theorem in [8] acknowledges the difficulties of solving the number-theoretical problems, but they provide some small values for which the theorem surely holds. However, the problem of determining if there are infinitely many such groups in the first three cases remains open.

Centralizers in power-cograph groups
Centralizers in power-cograph groups turn out to be quite restricted, which forms the basis of our analysis.
Lemma 4.1. Consider commuting group elements x, w ∈ G with |x| = p and |w| = q for two distinct primes p and q. If P ow(G) is a cograph, then all q-elements in C G (x) are contained in w . Furthermore, x is not a p-th power in G.
Proof. Assume that there is another element of order q in C G (x), z say, such that z ∈ w . Then is an induced P 4 in P ow(G) (we draw directions for clarity), contradicting the assumption that P ow(G) is a cograph. Thus, w is the unique subgroup of order q in C G (x). Again using the assumption that P ow(G) is a cograph, we note that, by the classification of nilpotent power-cograph groups (Theorem 3.5) and due to subgroup-closedness of powercograph groups, there are no elements of order Similarly, if x has a p-th root, say g p = x for a suitable g ∈ G, then Lemma 4.2. Consider commuting group elements x, w ∈ G with |x| = p and |w| = q for primes q < p. Assume P ow(G) is a cograph. Then x ∼ = C p and Q ∼ = w × C q n for some n ∈ N 0 . If n > 0 then Q/ w acts without fixed points on x , so q n divides p − 1.

Proof.
1. By Lemma 4.1, w is normal in C G (x). Since q < p, there is a unique normal Sylow p-subgroup P of C G (x) by the Sylow Theorems. Hence, C G (x) = w × P is nilpotent and thus P = x ∼ = C p via Theorem 3.5.
2. As in the first case, x C G (w) and then C G (w) = x ⋊ Q for a q-group Q by the Schur-Zassenhaus Theorem. Again using Lemma 4.1, we note that, up to powers, w is the unique q-element in C G (x). So C Q (x) = w and thus Q/ w ≤ Aut( x ) ∼ = C p−1 . By definition, w ∈ Z(Q) which implies that Q/Z(Q) is cyclic, hence Q is abelian. The lemma further gives us that w is not a q-th power and thus Q w × C q n .
Lemma 4.3. If P ow(G) is a cograph and {p, q} is an edge of π(G) with q < p, then the Sylow p-subgroups of G are isomorphic to C p and Sylow q-subgroups of G are of the form C q × C q n with n ∈ N 0 or have cyclic center.
Proof. By the previous lemma, there exists a p-element Consider a Sylow p-subgroup P of G containing x. Then 1 = Z(P ) ≤ C G (x), so Z(P ) = x . But then even P ≤ C G (x) holds, thus P = x . Now consider w ∈ G with |w| = q and assume p divides |C G (w)|. In this case the previous lemma still implies Lemma 4.4. Consider a semidirect product G ∼ = Q ⋊ C p where Q is a q-group for some prime q = p. If P ow(G) is a cograph and π(G) is not the null graph, then G ∼ = C q × C p .
Proof. The Sylow p-subgroups of G are isomorphic to C p and in particular, two p-elements in G always generate conjugate cyclic subgroups. Let x be an element of G of order p. Since π(G) is non-trivial, there must be some element w ∈ Q of order q with w ∈ C G (x). Assume there is some q-element y ∈ C G (w) \ w . Then yx ∈ C G (w) and due to the semidirect structure, p divides |yx|. But up to powers, x is the only p-element in C G (w), so (yx) d = x for an appropriate integer d. In particular it follows that yx ∈ C G (x) holds. On the other hand, by Lemma 4.1, w is the only q-element in C G (x), so x does not commute with y ∈ G \ w and thus, x does not commute with yx, a contradiction.
Hence, C G (w) = w × x and, as before, this implies Q = w due to Z(Q) = 1.
Lemma 4.5. Let G be a group and P ow(G) a cograph. Consider a connected component K of π(G). Then K is a star with center q, where q is the smallest prime in K.
Proof. Let q < p be primes that form an edge in π(G). Then C G (x) ∼ = C p × C q for some element x of order p and furthermore, x is a Sylow p-subgroup of G. Since Sylow psubgroups are all conjugate in G, it follows that all primes adjacent to p in π(G) already appear in |C G (x)| and then the lemma follows by repeating the argument for all primes in K \ {q}.
We can actually give a full characterization of power-cograph groups in terms of centralizer properties. We want to point out that [8, Theorem 3.1] contains a characterization of possible induced P 4 's in a finite group in terms of element orders and this is essentially the same argument we repeat in the following proof. We combine this with our restrictions on centralizers to turn it into a characterization in terms of centralizer structure.
Lemma 4.6. G is a power-cograph group if and only if for each prime p and for each element g ∈ G of order p the following hold: 2. if a prime q > p divides |C G (g)|, then there exists a normal subgroup N C G (g), N ∼ = C q , and C G (g)/N ∼ = g × C p n with n ∈ N 0 .
In particular, if C G (g) is not a p-group, then g is not a p-th power in G.
Proof. If G is a power-cograph group then centralizers of prime order elements behave as claimed according to Lemma 4.2. For the other direction, assume that Pow(G) is not a cograph. Then there exists an induced P 4 in Pow(G), given by (g 1 , g 2 , g 3 , g 4 ), say. Note that any two consecutive vertices cannot be of the same order, since elements generating the same cyclic group form a twin class in Pow(G). So without loss of generality we may assume that |g 2 | > |g 3 | holds. If |g 2 | is a power of some prime q, then either Condition 2 is violated or all of g 1 , . . . , g 3 are q-elements in g 2 , contradicting the assumption that the induced subgraph is a path. Thus, we may further assume that |g 2 | = pq with primes p = q. Then g 1 and g 3 need to be of order p and q, respectively. In particular, |g 4 | > |g 3 | holds. Say |g 3 | = q. If |g 4 | = pq, then g 3 commutes with two different elements of order p, contradicting our assumptions in any case. Otherwise |g 4 | is a power of q, but then g is a q-th power, again contradicting our assumptions.

Solvable groups
We focus on the case of solvable power-cograph groups first. Combining Lucido's Three Prime Lemma with our observations on centralizers in power-cograph-groups yields effective restrictions on the prime graph.
Lemma 5.1. If G is a solvable power-cograph group, then π(G) is an induced subgraph of either 2K 2 , the graph consisting of two disjoint edges, or P 3 , a path on 3 vertices.
Proof. By Lucido's Three Prime Lemma, π(G) has at most two connected components and if it does admit two components, then they can have at most two vertices each, since they are star graphs by Lemma 4.5 and there cannot be an induced coclique of size 3 by Lucido's Three Prime Lemma.
If π(G) is connected, then Lemma 4.5 implies that π(G) is a star with some central vertex p and π(G) − {p} can not have more than two vertices.
Lemma 5.2. Let G be solvable and assume Pow(G) is a cograph. If π(G) is connected then one of the following holds: 1. G is a group of prime power order.
2. π(G) consists of a single edge {q, p} with q < p. Then G ∼ = (C p ⋊ C q n ) × C q , and if n > 0, then q n divides p − 1 and C q n acts fixed-point-freely on the C p -part of Fit(G).
Conversely, all such groups are power-cograph groups.
Proof. By the previous lemma, π(G) is a star with center, say, q. Also all other primes dividing |G| are larger than q and divide |G| exactly once. Assume from now on that G is not a q-group.
Case 1: π(G) = {{q, p}}. Set F := Fit(G). Define F 2 ≤ G implicitly via F 2 /F := Fit(G/F ). If F is a p-group then F ∼ = C p is the unique subgroup of order p in G and selfcentralizing, contradicting the fact that π(G) is connected. Thus q divides |F |. Moreover the Fitting-group of a finite group is the direct product over all maximal normal prime power subgroups. Thus, if we assume F to be a q-group, we obtain that gcd(|F |, |F 2 |) = 1. By the Schur-Zassenhaus Theorem this implies that F 2 ∼ = F ⋊ C p is a semidirect product.
Since G is solvable, it holds that C G/F (F 2 /F ) ≤ F 2 /F and then for π(G) to be non-trivial, π(F 2 ) must be non-trivial. But then F 2 ∼ = C p × C q by Lemma 4.4, a contradiction.
In conclusion, it must hold that F is a nilpotent group whose order is divisible by pq, so F ∼ = C q × C p . Then it holds that G (C p ⋊ Q) × C q , where Q ≤ Aut(C p ) is a q-subgroup of the cyclic group Aut(C p ) acting fixed-point-freely on C p . The fact that F has a complement can be seen as follows. Let w ∈ F be an element of order q. Then w is centralized by each Sylow q-subgroup of G for q does not divide | Aut( w )|. By Lemma 4.2, the Sylow q-subgroups of C G (w) are isomorphic to w × C q n for some n.
Case 2: π(G) = {{q, p}, {q, r}}. As before, F cannot be of order p or r, since then the respective prime would have to be isolated in π(G) and F cannot be a q-group for the exact same reason we gave in Case 1 (note that the Fitting series of G does not contain a quotient of order pr). Then we have F ∼ = C q × C p (or F ∼ = C q × C r which is just a matter of renaming) and G/F ≤ Aut(C p × C q ). Now there is a unique subgroup of order q in F and thus elements of order r in G have to act fixed-point-freely on this subgroup (otherwise its centralizer would be divisible by q, p and r). But then r must divide q − 1, contradicting the fact that q is the minimal prime in its component of π(G). Thus, this case does in fact not occur.
The converse statement: Recall that groups of prime power order are always powercograph groups by Theorem 3.5. In the second case, the cograph property can be deduced from Lemma 4.6.
In particular, if G is a solvable power-cograph group and Fit(G) is of prime power order, then π(G) is disconnected. Here we should point out that in [8,Example 4.6.] the authors claim that the non-trivial semidirect product H 3 ⋊ C 2 is an example of a solvable power-cograph group whose socle is a 3-group (here, H 3 denotes the Heisenberg group of order 27). Since its prime graph is connected, our classification actually shows that the group is not a power-cograph group. Since the group contains centralizers isomorphic to C 3 × S 3 , the example already disagrees with Lemma 4.1. Lemma 5.3. Let G be solvable and assume Pow(G) is a cograph. If π(G) is disconnected, then one of the following holds: 1. the graph π(G) is the empty graph on two vertices. Here Fit(G) is a p-group and G/ Fit(G) is a generalized quaternion 2-group or a subgroup of C q k ⋊ C p m for a prime q = p and if m > 0 then C p m acts fixed-point-freely on C q k . In any case, Fit 2 (G)/ Fit(G) acts fixed-point-freely on Fit(G).
2. the graph π(G) consists of an edge {p, q} together with a third, isolated vertex r.
Then G is in one of two cases: (a) G = C pq ⋊ C r n with n ≥ 1 and C r n acts fixed-point-freely on C pq (in particular, r n divides q − 1 and p − 1).
is an r-group and Fit 2 (G) ∼ = C pq acts fixed-point-freely on Fit(G). Moreover, G/ Fit 2 (G) is either trivial or C r n acting as a fixed-point-free subgroup of Aut(C pq ).
3. the graph π(G) consists of two disjoint edges {p, q} and {r, s} where pq divides r − 1 and s − 1. Then G is of the form C rs ⋊ C pq with C pq acting fixed-point-freely on C rs . Conversely, all groups described above are power-cograph groups. Proof.
1. If π(G) is the empty graph, then Fit(G) has prime power order. Moreover, Fit 2 (G)/ Fit(G) is of prime power order coprime to | Fit(G)|. By [29, 10.5.6], the group Fit 2 (G)/ Fit(G) is cyclic or a generalized quaternion 2-group. In the first case, G = Fit 2 (G) or otherwise G is a 2-Frobenius group where G/ Fit 2 (G) is a cyclic group of prime power order acting fixed-point-freely on Fit 2 (G)/ Fit(G). In the second case, Fit 2 (G)/ Fit(G) contains a unique element of order 2 so it does not allow fixed-point-free extensions, i.e., G = Fit 2 (G). This leaves us with two more options for π(G) according to Lemma 5.1. Otherwise, Fit(G) is either a p-group or a q-group. Without loss of generality, assume that Fit(G) is a q-group. Then there is an element of order pq in G and together with Fit(G) it spans a semidirect product of a q-group with C p , having nonempty prime graph. Thus, Lemma 4.4 implies that Fit(G) is trivial, a contradiction.
We established that the only case left to consider is the case where Fit(G) is an r-group. Then Fit 2 (G)/ Fit(G) ∼ = C p × C q , otherwise we can repeat the argument from above, since G/ Fit(G) now has to contain elements of order pq. If it holds that G = Fit 2 (G), we are done. Otherwise G/ Fit 2 (G) acts on Fit 2 (G)/ Fit(G) ∼ = C pq as a subgroup of Aut(C pq ), where the latter is an abelian group. Without loss of generality assume that q < p, so p divides |G| exactly once by Lemma 4.3. If G/ Fit 2 (G) contains elements of order q, then by Lemma 4.2, G/ Fit(G) contains a subgroup isomorphic to C q × C q , acting on Fit(G) without fixed-points. This is not possible, see [29, 10.5.3]. Thus, G/ Fit 2 (G) is an abelian r-group acting on Fit 2 (G)/ Fit(G) as a fixed-point-subgroup of Aut(C p ) and Aut(C q ) simultaneously.
3. Case π(G) = ({p, q, r, s}, {{p, q}, {r, s}}): If Fit(G) ∼ = C rs then G/ Fit(G) is abelian with at least two prime divisors p and q, so G/ Fit(G) ∼ = C pq and this group must act on Fit(G) without fixed points. Otherwise Fit(G) must be of prime power order (since it is a nilpotent power-cograph group) and then G cannot be a Frobeniusgroup, where π(Fit(G)) would form a connected component in π(G). Thus, G is 2-Frobenius with corresponding normal subgroups K N G (cf. Lemma 2.5). Then π(N/K) forms a component in π(G) which must be of size two by the assumptions on π(G). Without loss of generality let N/K ∼ = C rs and let K be a q-group. By definition, G/K is a semidirect product with normal subgroup N/K so we may choose some subgroup H ≤ G such that K ≤ H and H/K is a complement of N/K in G/K. If both p and q divide |H/K|, then H/K ∼ = C pq , since H/K acts on N/K as an automorphism subgroup and Aut(N/K) is abelian. In this case, H ∼ = (K.C q )⋊C p with non-trivial prime graph, implying K = {1} via Lemma 4.4, a contradiction. Otherwise, H/K is a p-group (because p must divide |H/K| at least once) and then K ∼ = C q , again using Lemma 4.4 and the fact that some element of order q must commute with an element of order p. But then C G (K) ∼ = C p × C q is normal in G, contradicting the fact that K = Fit(G) is a q-group. Finally, we point out that all groups appearing in the lemma's statement are indeed power-cograph groups according to Lemma 4.6.
While it would be even better to have a list of possible Fitting-subgroups in Cases 1) and 2b), the restrictions we give precisely describe the structure of G. It remains the question of which p-groups admit fixed-point-free automorphisms of prime order. For abelian Fitting-subgroups the question can be answered in full generality [26], so we obtain a classification under this extra assumption. For the general case, there is an extensive body of research concerned with bounding the Fitting-length of groups admitting fixedpoint-free automorphisms of prime order, see for example [17,37]. In particular, they are always nilpotent by a result of Thompson [36], but no classification is available.

Non-solvable groups
To deal with non-solvable groups we first analyze the socle of a power-cograph group to split up the classification task into smaller sub-problems. Since a high number of direct factors in Soc(G) implies the existence of relatively large element centralizers, the socle of a power-cograph turns out to be quite restricted. Lemma 6.1. Let G be a non-solvable group whose power graph is a cograph. Set S := Soc(G). Then one of the following holds: 1. S is a non-abelian simple group and S G ≤ Aut(S), so G is almost simple.
2. S ∼ = C n p for some n ∈ N and G/C ≤GL n (F p ), where C := C G (S) is a p-group. Proof. We can write S = A × T 1 × · · · × T m where A is an abelian group and the T i are non-abelian simple groups. If m ≥ 1 and A is non-trivial, then there exists a ∈ A such that, without loss of generality, T 1 ∈ C G (a) holds. Since |T 1 | is divisible by at least three primes, the order of C G (a) is divisible by at least three primes but by Lemma 4.2, this is impossible. An analogous argument applies if m > 1 holds, even if A is trivial. Thus S is either an abelian group or a non-abelian simple group. In the second case, G is almost simple. In the first case A is nilpotent and so it is either a p-group or isomorphic to C p × C q for primes p = q. Assume first that S ∼ = C p × C q holds. Then G/C G (S) ≤ Aut(S) and since the latter is abelian, C G (S) must have a non-abelian simple factor, contradicting Lemma 4.2. Thus, S ∼ = C n p for some prime p and it follows that C G (S) is also a p-group. Otherwise there is a q-element centralizing S for some prime q = p, forcing n to be 1 according to Theorem 3.5. In the same way as before, this contradicts Lemma 4.2 through G being non-solvable.

The almost simple case
Recall that the class of power-cograph groups is closed under taking subgroups. This means whenever G is an almost simple power-cograph group, the socle Soc(G) is a simple power-cograph group. We refer to Theorem 3.6 for a characterization of non-abelian finite simple groups that are power-cograph groups.
Lemma 6.2. If G is an almost simple power-cograph group, then G is either simple or G = PGL 2 (q) for some prime power q or G = M 10 .
Proof. Let Soc(G) be a non-abelian simple group so that G is an almost simple group. Note that Soc(G) must be a power-cograph group due to subgroup-closedness. Due to Theorem 3.6, it suffices to check automorphisms of PSL 3 (4), PSL 2 (q) and Sz(q).
We can immediately exclude the first case via computations we performed in GAP [13].
Extensions of PSL 2 (p f ) by field automorphisms, p odd: Suppose G = PSL 2 (p f ) α where p is an odd prime and α is a field automorphism whose order divides f . Then α is a field automorphism of F f p which acts on PSL 2 (q) by acting on the respective matrix representatives in SL 2 (q) entry-wise.
Case f even: Then p 2 − 1 divides p f − 1 and thus there exist elements of order (p − 1)(p + 1)/2 in PSL 2 (p f ). If p = 3, then (p − 1)(p + 1) can neither be a product of two primes nor a prime power (since p > 3 is prime) so PSL 2 (p f ) is not a power-cograph group by Theorem 3.6. If p = 3 and f > 2, then either 40 = (3 4 − 1)/2 divides (p f − 1)/2 or 3 2g − 1 divides p f − 1 for some odd number g. If the first case holds, then again PSL 2 (p f ) is not a power-cograph group. In the second case, (3 g − 1)(3 g + 1) divides p f − 1. Then (3 g + 1) is divisible by 4 since g is odd, and 3 g − 1 is not a power of 2, so (p f − 1)/2 is divisible by 4m, where m is an odd number. As in the first case, PSL 2 (p f ) is not a power-cograph group. In conclusion, either f = 2 and p = 3 or f is odd.
In the case f = 2 and p = 3, it holds that G ∼ = S 6 , and G is not a power-cograph. Case f odd: Since α fixes elements of the base field F p , in particular all matrices defined over F p are fixed. Therefore, PSL 2 (p) C G (α). Then if p > 3, the order of C G (α) must be divisible by at least three primes as the group PSL 2 (p) is a nonabelian simple group, and by Lemma 4.2 this is impossible. If p = 3, then PSL(2, 3) ∼ = A 4 is centralized by α. If C G (α) fulfills the necessary conditions in Lemma 4.2, then the order of α must be 2, but we assumed f to be odd here.
Extensions of PSL 2 (p f ) by diagonal-field automorphisms, p odd: Here G = PSL 2 (p f ) α where p is an odd prime and α = δσ is a product of a diagonal automorphism δ and a non-trivial field automorphism σ. If |σ| is odd, then G contains a subgroup PSL(2, p f ) σ ′ for some field automorphism σ ′ of odd order, and the argument from above implies that G is not a power-cograph group. So |σ| is even. If f is even, the arguments from above leave us with the case of PSL (2,9) α , where α has order 2. Then G ∼ = M 10 which is an EPPO-group, and in particular, it has a power-cograph.
Extensions of PSL 2 (2 f ) by field automorphisms Suppose now G = PSL 2 (q) α where q is a power of 2. Then every outer automorphism is a field automorphism, and each field automorphism fixes at least a copy of PSL 2 (2) ∼ = S 3 . This can only happen if the field automorphism is of order 2 and has no fixed field larger than F 2 , which implies that q = 4. But PSL 2 (4) ∼ = PSL 2 (5) has an automorphism group isomorphic to PGL 2 (5).

Extensions of Suzuki groups by field automorphisms:
We use a similar argument for the Suzuki group case as in the PSL 2 (q) case. Let G = Sz(2 f ) α with α a field automorphism whose order divides f , where f is odd and f ≥ 3. Then α fixes a subgroup isomorphic to Sz(2) of order 20, and again the result follows by Lemma 4.2.
In conclusion, the only remaining cases are groups of the form G = PGL 2 (q) for an appropriate prime power q. Lemma 6.3. Let PSL 2 (q) G ≤ PGL 2 (q) and assume G is non-solvable. Then G is a power-cograph group if and only if one of the following holds: 1. q > 3 and G = PSL 2 (q) 2. q is 5, 7, or 9.
Proof. If q is even then PSL 2 (q) = PGL 2 (q), thus assume that q is odd. Then PSL 2 (q) has index 2 in PGL 2 (q), so if G is not equal to PSL 2 (q) then G = PGL 2 (q) holds. In the latter case, G has cyclic subgroups of order q − 1 and q + 1 and their orders must be prime powers or a product of two distinct primes for Pow(G) to be a cograph. But one of q − 1 and q + 1 is divisible by 4, so one of them is a 2-power and the other is 2r with a prime r > 2.
We could be dealing with the unique solution to the Catalan Conjecture, i.e., q = 9 and q − 1 = 8. Since q = 9 gives q + 1 = 10, PGL 2 (9) is indeed power-cograph. Furthermore, q > 3 implies 2 < q − 1, q + 1, so this is actually the only possible case if q is not prime.
We are left with two cases, since q is a prime of the form 2 a ± 1 now. If q = 2 a + 1 then q + 1 = 2(2 a−1 + 1), so 2 a−1 + 1 must be prime as well. A well-known result (see for example [28,Page 1]) states that both a and a − 1 must be powers of 2 (including the possibility 1 = 2 0 ) which is only possible if a = 2. This gives us PGL 2 (5), which is again power-cograph. If q = 2 a −1 then q −1 = 2(2 a−1 −1) and so 2 a−1 −1 is an odd prime. It is well-known that then both a and a − 1 have to be primes (see for example [28, Appendix B, Theorem B.1]), so a = 3. Indeed PGL 2 (7) is also a power-cograph group.

The abelian socle case
Recall that 2-Frobenius groups are always solvable. We observe the following additional restriction for non-solvable groups. Lemma 6.4. If G is a non-solvable power-cograph group, then G is not a Frobenius group.
Proof. Let G = K ⋊ H be a non-solvable Frobenius group. Then the Frobenius complement H (and thus G) contains a subgroup isomorphic to SL 2 (5), which has non-trivial center and more then 3 primes dividing its order. In particular, SL 2 (5) is not a powercograph group according to Lemma 4.2.
Lemma 6.5. If G is a non-solvable power-cograph group with an abelian socle, then 2 is isolated in π(G). In particular, π(G) is disconnected.
Proof. Let S = Soc(G). If S = C p , then consider the action of G on S via conjugation. This induces φ : G → Aut(S) with kernel C G (S). Since G is non-solvable, either ker φ or Im φ contains a non-abelian simple composition factor. However, Aut(S) is abelian and C G (S) cannot contain this factor by Lemma 4.2, so this is a contradiction. Now either π(G) is disconnected and then all prime divisors of | Soc(G)| must be in the component of 2 (Lemma 2.5) or π(G) forms a star with center 2 (Lemma 4.5). In any case, odd primes divide | Soc(G)| at most once and since we can rule out cyclic socles by non-solvability of G, we can assume that S = F d 2 with d ≥ 1. Assume that 2 is not isolated in π(G), so there exists g ∈ G such that |g| = p = 2 with |C G (g)| even. Suppose w ∈ C G (g) such that w has order 2.
Let H be a Sylow 2-subgroup of G with w ∈ H. Since S G and S is a 2-group, we have S ≤ O 2 (G), where O 2 (G) denotes the largest normal 2-subgroup of G. Note that O 2 (G) is the intersection of all Sylow 2-subgroups of G. Then S H and S ∩ Z(H) is non-trivial because H is a Sylow 2-subgroup and any normal subgroup intersects the center non-trivially. So there exists z ∈ S ∩ Z(H) such that |z| = 2 and z ∈ C G (w). Now suppose z ∈ C G (g). Since z centralizes w and w centralizes g, we have that z normalizes g C G (w). Furthermore, zgz −1 = zgz = g −1 because z commutes with w and it has order 2. Now g normalizes S so gzg −1 ∈ S \ z because g and z do not commute. Thus z, gzg −1 ∼ = V 4 since it is a subgroup of F d 2 and it is not cyclic. However, zgzg −1 = g −2 is of order p and an element of V 4 , which is a contradiction. Thus z ∈ C G (g) and g ∈ C G (z). By Lemma 4.2, C G (z) = g ⋊ Q, where Q is a q-group of rank at most 2. As S ⊆ C G (z), S ∼ = F 2 or S ∼ = F 2 2 . Then G/C G (S) ≤ Aut(S) (as above, by Lemma 4.2, C G (S) cannot have non-abelian simple composition factors) must have a non-abelian simple composition factor, which is impossible due to GL 2 (2) ∼ = S 3 . Lemma 6.6. If G is a non-solvable power-cograph group, then G has a normal subgroup N that is either trivial or a 2-group, such that in any case G/N is almost simple. Moreover, G/N is a power-cograph group.
Proof. If N = 1, then the statement follows from the definition, so assume otherwise. Combining Lemma 2.5 and Lemma 6.5, we may assume that G is an extension of a nilpotent π 1 -group by a simple group by a π 1 -group, where π 1 denotes the component containing 2 in π(G). We showed earlier that 2 is isolated in π(G) if Soc(G) is a 2-group, and so G/O 2 (G) ∼ = T.H, where H is a 2-group and T is simple.
To see that Pow(G/N) is a cograph, recall Lemma 3.4. Since 2 is isolated in π(G), any induced P 4 in Pow(G/N) would contain odd-order elements only and also the relevant centralizer defined in Lemma 3.4 must be of odd order. Thus, all induced P 4 's in Pow(G/N) would lift to G.
We refer to Theorem 3.6 once again to deduce that T may only be one of PSL 3 (4), PSL 2 (q) where q is a prime power, or Sz(q) where q is a power of 2.
Lemma 6.7. Let G be a non-solvable power-cograph group with abelian socle, then S := Soc(G) ∼ = F d 2 with d ≥ 1. Moreover, G/O 2 (G) has a non-abelian simple socle T and one of the following holds: 1. T ∼ = PSL 2 (2 n ) with n ≥ 2. Then S = O 2 (G) and each minimal normal subgroup of G is isomorphic to the natural F 2 n [SL 2 (2 n )]-module as a G/S-module. (2 2e+1 ) with e ≥ 2. Then S = O 2 (G) and each minimal normal subgroup of G is isomorphic to the natural [F 2 2e+1 Sz(2 2e+1 )]-module as a G/S-module.

T ∼ = Sz
In both cases G is a power-cograph group if and only if T is a simple power-cograph group where 2 is isolated in π(G).
Proof. By the previous lemma and Lemma 6.1, G/O 2 (G) has a non-abelian simple socle whose power graph is a cograph. According to [8,Theorem 1.3], T := Soc(G/O 2 (G)) is either PSL 3 (4), PSL 2 (q) or Sz(q) for appropriate values of q. We may further assume that 2 is isolated in π(G) by Lemma 6.5 and in particular, all odd order elements in G must act on S fixed-point-freely. We checked with GAP [13] that we can exclude PSL 3 (4) by explicitly computing odd order elements with fixed-points in all irreducible representations of PSL 3 (4) over F 2 .
Furthermore, by [32,Proposition 4.1] and [25, Theorem, Remark 1], it must be the case that O 2 (G) is elementary abelian and forms a direct sum of copies of the natural SL 2 (q)-module or the natural Sz(q)-module, respectively.
We now prove that under these assumptions, H := S.T ≤ G is a power-cograph group whenever T is. The general approach is to apply Lemma 4.6.
Consider commuting elements of distinct prime orders in H, say x and y . Then C H (x)S/S is either a p-group or isomorphic to C p × C q for appropriate primes p and q, since this is the case for T whenever T is a power-cograph group, see Theorem 3.6. In the first case, one of x and y has order 2 and since 2 is isolated in T the respective element, y say, must be contained in S.
In the second case, C H (x) is either C p × C q and thus admissible according to Lemma 4.6, or there is some non-trivial element in S that commutes with an element of order p = 2. In any case, the only obstruction to C H (x) being admissible is the existence of an element t ∈ T of prime order p = 2, such that C S (t) = {1} holds. We argue that such elements cannot exist.

Acknowledgements
We thank an anonymous referee for pointing out a gap in a previous proof of Lemma 6.2 which led to the inclusion of the group M 10 into Theorem 1.1.