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Finite groups isospectral to simple groups

The spectrum of a finite group is the set of element orders of this group. The main goal of this paper is to survey results concerning recognition of finite simple groups by spectrum, in particular, to list all finite simple groups for which the recognition problem is solved.


Introduction
The elementary assertion that groups of exponent 2 are abelian and one of the most complicated theorems in the finite group theory on solvability of groups of odd order are similar in the following sense. In both cases, arithmetic, i.e., expressed by numeric parameters, properties of a group allow to conclude on its structure. The strongest conclusion that can be made in this direction is that a group is uniquely, up to isomorphism, determined by a set of numeric parameters. In such case the group is said to be recognizable by this set.
The recognition of finite simple groups -the building blocks of finite group theory -are definitely of prime interest. According to their classification (CFSG), finite simple groups are exactly (i) the groups of prime order; (ii) the alternating groups of degree at least 5; (iii) the simple classical groups; (iv) the simple exceptional groups of Lie type; (v) the 26 sporadic groups.
In 1987, in his letter to J. Thompson, W. Shi conjectured that every finite simple group is recognizable in the class of finite groups by its order and the set of element orders. In his answer, J. Thompson highly appreciated this conjecture and put forward another one: every finite nonabelian simple group is recognizable in the class of finite groups with trivial center by the set of sizes of conjugacy classes. Later, A. S. Kondrat'ev added the questions about the validity of these conjectures to the Kourovka Notebook [52,Problems 12.38,12.39].
As a result of efforts by numerous mathematicians started in [97] and [18], the validity of both conjectures was eventually established: Shi's conjecture -in 2009 [119], and Thompson's conjecture -in 2019 [26]. Together with [53,Corollary 5.2], the validity of Shi's conjecture implies that a finite simple group and an arbitrary finite group having the same Burnside ring are isomorphic, i.e., every finite simple group G is recognizable in the class of finite groups by G-sets (Yoshida's problem [129,Problem 2] for simple groups).
In Shi's conjecture, a recognition is based on the order and the set of element orders. What happens if we leave only the latter of these parameters -the set of element orders? Following [1], we refer to this set as the spectrum of a group G and denote it by ω(G), while 1 groups with the same spectrum are said to be isospectral. Observe that another notation for the spectrum is π e (G) (see, for example, [105]).
In the middle of the 1980s W. Shi [91,92] discovered that P SL 2 (7) and Alt 5 can be characterized in the class of finite groups solely by the spectrum, and these results opened a wide way for investigations of recognizability of groups by spectrum. Though a finite group G isospectral to a nonabelian simple group L is not necessarily isomorphic to L, it is very close to L in the vast majority of cases. Namely, if L is alternating of sufficiently large degree or classical of sufficiently large dimension, then G is isomorphic to a group squeezed between Inn L and Aut L, the groups of inner and of all automorphisms of L respectively. This property of 'sufficiently large' simple groups was conjectured by V. D.  and eventually proved in 2015 (see [42] and the references therein).
We denote the number of pairwise nonisomorphic finite groups isospectral to a group G by h(G). So G is recognizable (by spectrum) if h(G) = 1. A group G is said to be almost recognizable if h(G) is finite, and unrecognizable if h(G) = ∞ [103]. We say that the recognition problem is solved for G if the number h(G) is known, and if it is finite, then all the groups isospectral to G are described. The main goal of this paper is to survey known results on this problem. The previous surveys can be found in [39,74,75,78,100,104,105] and partially in the introduction in [42].
Since the study of the recognition problem has naturally gone in parallel with the study of spectra, it is worth noting that the spectra of all finite nonabelian simple groups are known. Clearly, the spectra of alternating groups are easy to describe, and the spectra of sporadic groups are known due to [20]. The spectra of linear and unitary groups are found in [14], and those of symplectic and orthogonal in [15]. For exceptional groups, the task was completed in [16] with describing the spectra of E 8 (q).
The structure of the paper is as follows. Section 2 is concerned with finite simple groups for which the recognition problem is solved (Theorem 2.1), and the groups themselves are listed in Appendix. In Section 3, we state some general properties of finite groups isospectral to finite simple groups and discuss finite simple groups for which the recognition problem is not solved. Finally, we discuss some related problems and open questions in Section 4.

2.
Simple groups with solved recognition problem 2.1. The main theorem. The finite nonabelian simple groups for which the recognition problem is solved are listed in Tables 1-9 in Appendix. The main result of the section is Theorem 2.1 which describes finite groups isospectral to L for every simple group L listed in the tables and having h(L) < ∞.
We denote the alternating and symmetric groups of degree n by Alt n and Sym n respectively. In notation of the sporadic groups and groups of Lie type we follow the 'Atlas of finite groups' [20], with the exception that we write 2 B 2 (q), 2 G 2 (q) and 2 F 4 (q) for the Suzuki-Ree groups. Also we use the abbreviations L ± n (q) and E ± 6 (q), where L + n (q) = L n (q), L − n (q) = U n (q), E + 6 (q) = E 6 (q), E − 6 (q) = 2 E 6 (q). As usual, we identify L with Inn L and write Out L for the quotient Aut L/ Inn L, the outer automorphism group of L. Proof. It should be noted that in this proof, for the sake of brevity, we generally refer not to all relevant papers but only to the papers in which some final steps were made or some remaining cases were settled. We will give a more detailed exposition in Section 3.
, then see [107]. Thus we may assume that L is a classical group in odd characteristic with h(L) < ∞ and . Let G be a finite group isospectral to L. First, we need to prove that G is an almost simple group with socle L, that is, L ≤ G ≤ Aut L.
where n 4 is a power of 2, then by [116,Theorems 1,2], it follows that G has a unique nonabelian composition factor and this factor is isomorphic to L. The same is true when L is one of the groups in Table 7 (see [47] for U n (3), [49] for S 2n (3), [90] for O 2n+1 (3), and [7,8,48,56,57] for O ± 2n (q)). Now we apply [32, Theorem 1.1] to conclude that the solvable radical of G is trivial, and so G is an almost simple group with socle L, as required.
It remains to determine almost simple groups with socle L isospectral to L. Such groups are described in [34] for linear or unitary socle, in [33] for symplectic or odd-dimensional orthogonal socle, and in [35] for even-dimensional orthogonal socle.
2.2. Number-theoretical notation and outer automorphisms used in the tables. Given a positive integer n, we write π(n) and τ (n) for the set of prime divisors of n and the number of all divisors of n respectively. For a prime r, we write (n) r for the r-part of n, that is, the highest power of r dividing n; and (n) r ′ denotes the r ′ -part of n, that is, the ratio n/(n) r . More generally, if m is a positive integer, then (n) m is equal to the product of (n) r over all r ∈ π(m) and (n) m ′ = n/(n) m . The greatest common divisor and least common multiple of integers n 1 , n 2 , . . . , n s are denoted by (n 1 , n 2 , . . . , n s ) and [n 1 , n 2 , . . . , n s ] respectively. For simplicity, we write (n 1 , n 2 , . . . , n s ) r instead of ((n 1 , n 2 , . . . , n s )) r for the r-part of (n 1 , n 2 , . . . , n s ). Also for ε ∈ {+, −}, we write ε in place of ε1 in arithmetic expressions.
Recall that a Mersenne prime r has the form r = 2 k − 1 for some k. Following [133], we say that a Mersenne prime r is special if r 2 − r + 1 is also a prime. For example, 3 and 7 are special Mersenne primes (in fact, these are the only known special Mersenne primes).
We write diag(a 1 , a 2 , . . . , a n ) for the matrix A = (a ij ) of size n × n with a ii = a i for all i and a ij = 0 otherwise. We write antidiag(a 1 , a 2 , . . . , a n ) for the matrix A = (a ij ) of size n × n with a i,n−i+1 = a i for all i and a ij = 0 otherwise. The transpose of A is denoted by A ⊤ .
Let L be a group of Lie type. We are going to introduce notation for outer automorphisms of L involved in the structure of groups isospectral to L. Since the Suzuki-Ree groups are recognizable by spectrum, in the following definition of automorphisms, we assume that L is not a Suzuki-Ree group.
Let L be a group over a field of characteristic p and order q. It is well known that we can identify L with a group of the form O p ′ (L σ )/Z(O p ′ (L σ )), where L is a suitable simple algebraic group over the algebraic closure F of the field of order p, σ is a suitable endomorphism of L, and L σ = C L (σ). The map x α (t) → x α (t p ), where x α (t) are root elements of L, induces an endomorphism of L denoted by ϕ p (see [22,Theorem 1.15.4(a)]). If q = p m , then ϕ q stands for (ϕ p ) m .
We may assume that σ = ϕ q γ 0 , where γ 0 is some graph automorphism. We take γ 0 to be the graph automorphism induced by a suitable symmetry of the Dynkin diagram of L as in [22,Theorem 1.15.2(a)] with the following two exceptions. If L is L ε n (q) or O ε 2n (q) with q odd, we need a specific γ 0 and we define it as follows.
Let p be odd. If L = L ε n (q) with n 3, then we assume that L = SL n (F ) and define γ 0 to be the inverse-transpose automorphism g → g −⊤ of L. If L = O ε 2n (q), then we assume that L = SO(V , f ), where V is a vector space of dimension 2n over F and f is a nondegenerate quadratic form. We can choose a basis of V so that the matrix of f with respect to this basis is antidiag(1, 1, . . . , 1). Denote by γ 0 the linear transformation of V interchanging the first and last basis vectors and fixing all others. Now we define ϕ and γ to be the images in Out L of the automorphisms of L induced by ϕ p and γ 0 respectively. Also, if L = L ε n (q) and λ is a primitive (q − ε)th root of unity, then δ is the image in Out L of the diagonal automorphism of L induced by conjugation by diag(λ, 1, 1, . . . , 1).
If α ∈ Out L, then L. α denotes the full preimage of α in Aut L.  (except for Table 9), we introduce the numbers and automorphisms used in the body of the table. The first two columns provide restrictions on L. The column 'h(L)' specifies the number of groups isospectral to L. If h(L) = ∞, then the column 'Note' provides the reference to the paper where this fact was proved. If h(L) is finite but not 1 and L = S 6 (2), , then the column 'Note' specifies the subset Θ of Out L whose meaning is explained in , then h(L) = 2 and the column 'Note' specifies the only finite group isospectral but not isomorphic to L. Finally, if h(L) = 1, then the corresponding entry of this column is empty. Also Tables 1 and 2 contain the references to the following Lemma 2.2 which gives explicit arithmetic conditions for some almost simple groups to be not isospectral. These conditions are too voluminous to be conveniently included in the tables. (i) n = p t−1 + 2 with t 1 and q ≡ −ε (mod 4); (ii) n = 2 t + 1 with t 1 and (n, q − ε) > 1;  Let p be odd.
If n − 1 is a power of p, then p = 3. Since n − 2 = 2 s , we need to look at the value of (b) 2 = (m) 2 . It follows that h(L) = 1 if (m) 2 2 and h(L) = 2 otherwise. In the latter case, ω(L) = ω(L. χγδ ), where χ is a field automorphism of order 2.
Let p = 2, 3. Suppose that m is odd. Then b is odd, and applying Lemma 2.2, we conclude that ω(L. γ ) = ω(L) if and only if q ≡ 1 (mod 4) or q ≡ 1 (mod 7), in other words, when Now suppose that m is even. The condition n = p s +2 u +1 holds if and only if p = 5, 11, 19 or 23. In these cases, . For example, if q = 5 42 , then b = 14 and h(L) = 6, with the groups isospectral to L being L, L. ξ , L. ξγ , L. ψ , L. ψξ , and L. ψξγ , where ψ and ξ are field automorphisms of orders 7 and 2, respectively. Observe that, contrary to the previous cases, here we have two maximal nonisomorphic subgroups of Aut L among the groups isospectral to L.
3. The structure of groups isospectral to simple groups 3.1. General results. We need a definition of the prime graph (or the Gruenberg-Kegel graph) GK(G) of a finite group G: the vertex set of this graph is π(G) and primes r, s ∈ π(G) are adjacent if and only if r = s and rs ∈ ω(G). Recall that a coclique is a graph in which all the vertices are nonadjacent. The maximal size of a coclique in GK(G) is denoted by t(G) and for r ∈ π(G), the maximal size of a coclique in GK(G) containing r is denoted by t(r, G).
Observe that the prime graph of a solvable group G contains no cocliques of size 3, and so t(G) 2. This is a direct consequence of G. Higman's result [51, Theorem 1], but as an independent statement, it is due to M. S. Lucido [62,Proposition 1].
The connected components of the prime graphs of finite simple groups were described by J. S. Williams [126] and A. S. Kondrat'ev [54]. Explicit criteria of adjacency in these graphs were found by A. V. Vasil'ev and E. P. Vdovin [124] (see also [125,Section 4] and [49, Section 2] for corrections to [124]). One of important corollaries of Vasil'ev and Vdovin's result is that t(2, L) 2 for all finite simple groups L except some Alt n with n 27 [124, Theorem 7.1]. The importance of this fact for the recognition problem is explained by the following result of A. V. Vasil'ev. In 2013 I. B. Gorshkov [24] proved that a finite group G isospectral to Alt n with n 5 and n = 6, 10 is isomorphic to Alt n and, in particular, has a unique nonabelian composition factor. Also by [63], [73,Proposition 3], [3, Example 2], [109], and [136], the simple groups isospectral to solvable groups are exactly L 3 (3), U 3 (3), and S 4 (3). Thus we have the following theorem. In what follows in this section we assume that G is a nonsolvable group isospectral to a simple group. By Theorem 3.2, it follows that G has the normal series where K is the solvable radical of G, that is, the largest normal solvable subgroup of G, S = H/K is a nonabelian simple group, and G/K is a subgroup of Aut S. By the validity of Mazurov's conjecture, for 'sufficiently large' simple groups, we have S ≃ L and K = 1.
But what is known about S, K, and G/H, in general? If L has a disconnected prime graph, then it follows from the Gruenberg-Kegel theorem [126, Theorem A] and Thompson's theorem on nilpotency of Frobenius kernel that K is nilpotent. It turns out that K is nilpotent for all L other than Alt 10 . 128]). Let L be a finite nonabelian simple group and suppose that G is a nonsolvable finite group isospectral to L. If L = Alt 10 , then the solvable radical of G is nilpotent.
By [72], the group Alt 10 is isospectral to a group of the form (7 4 × 3 12 ) : (2.Sym 5 ) whose solvable radical is a Frobenius group with complement of order 2, so Alt 10 is indeed an exception to Theorem 3.3. Another exceptional feature of this example is that the solvable radical here is a 3-primary group. Also by [68], there is a nonsolvable group with biprimary solvable radical isospectral to S 4 (3). We suggest that these are the only examples of a non-primary solvable radical. Recall that G/H can be identified with a subgroup of Out S, and so if S is sporadic, or Alt n with n = 6, or a Suzuki-Ree group, then G/H is cyclic. Recently, it was proved that G/H is always cyclic.
Theorem 3.5 ([43, Theorem 1]). Let L be a finite nonabelian simple group and suppose that G is a nonsolvable finite group isospectral to L. If 1 ≤ K < H ≤ G is the series as in (3.1), then G/H is cyclic.
3.2. The groups K and G/H in the case S ≃ L. The structure of K and G/H is best studied when S ≃ L. We refer to a finite group that can be homomorphically mapped onto L as a cover of L. If S ≃ L, then not only G but also both H and G/K in (3.1) are isospectral to L. So we have, on the one hand, a cover of L isospectral to L and, on the other hand, an almost simple group with socle L isospectral to L.
The first examples of groups not recognizable among covers, namely U 3 (3) and U 3 (7), were found by V. D. Mazurov in [72]. A. V. Zavarnitsine [133] generalized these examples to the assertion that U 3 (q) is recognizable among covers if and only if q is not a special Mersenne prime (recall from Section 2 that a Mersenne prime r is special if r 2 − r + 1 is a prime too). The covers of other unitary and linear groups were settled in [31,40,132,135,138], of symplectic and orthogonal groups in [31,32]. The groups 3 D 4 (q) with q = 2, F 4 (q), E ε 6 (q) and E 7 (q) are recognizable among covers by [32,123]. Finally, 3 D 4 (2) is not recognizable among covers by [76]. The following theorem summarizes the results on recognition among covers.
Theorem 3.6. Let L be a finite nonabelian simple group and G is a finite group such that G/K ≃ L for some nontrivial normal subgroup K of G. If L = U 3 (q), where q is a special Mersenne prime, and L = U 5 (2), 3 D 4 (2), then ω(G) = ω(L).
'Atlas of finite groups' provides several examples of almost simple groups isospectral to their socles, for example, L 3 (5) is isospectral to its extension by the graph automorphism. It turned out that this situation is rather typical for simple groups of Lie type, in other words, simple groups of Lie type are mostly not recognizable among automorphic extensions. This leads to the problem of describing almost simple groups of Lie type isospectral to its socle. A. V. Zavarnitsine [131,133] solved this problem for L 3 (q) and U 3 (q) and provided the first examples showing that there can be arbitrarily many almost simple groups with the same socle isospectral to this socle. Later the problem was solved for the classical groups in characteristic 2 [29,38,140] and the groups 3 D 4 (q), F 4 (q), E ε 6 (q), E 7 (q) [45,141]. The final steps relating to classical groups in odd characteristic were done by M. A. Grechkoseeva [33][34][35], with substantial use of the description of the spectra of these groups given by A. A. Buturlakin [14,15]. As a result of all these contributions, we have the following theorem. 3.3. Quasirecognizability. Discussing the factor S = H/K, it is convenient to use the notion of quasirecognizability introduced by Kondrat'ev [9]. A finite simple group L is said to be quasirecognizable by spectrum if every finite group G isospectral to L has a unique nonabelian composition factor and this factor is isomorphic to L. So in terms of (3.1), L is not quasirecognizable if there exists some G isospectral to L in which S ≃ L.
Suppose that q is odd. The groups L 2 (q), L 3 (q) and U 3 (q), except for L 2 (9), L 3 (3) and U 3 (3), are quasirecognizable. The groups S 4 (q) are quasirecognizable if and only if q = 3 2k+1 > 3. For the classical groups in larger dimensions, the study of S fell naturally into four cases according as S is sporadic, or alternating, or of Lie type in characteristic p, or of Lie type in characteristic not p. The first three cases are covered in [42,111,120,121]. The final result concerning these three cases is the following.  In fact, the recognition problem for S 8 (7 m ) is equivalent to Problem 3.9. Indeed, suppose that we fix m and let L = S 8 (7 m ). By [35,Lemma 4.3], if G is isospectral to L then G is either an almost simple group with socle L, in which case applying Theorem 3.7 we see that G = L, or an extension of a nontrivial 7-group by an almost simple group with socle O − 8 (7 m ). So if the group G of Problem 3.9 does not exist, then h(L) = 1, and if it exists, then h(L) = ∞ by Theorem 3.12 below.
The last case when S is a group of Lie type in characteristic not p is the most difficult, and for a long time this case could be excluded only in some special circumstances, for example, when the prime graph of L is disconnected. In 2015 A. V. Vasil'ev [115] made a breakthrough in this direction eliminating this case for all classical groups of dimension at least 62 (more precisely, for all classical groups L with t(L) 23). Later A. S. Staroletov [108] lowered the bound to 38. More precisely, Staroletov's bound depends on the type of L and is equal to 27 for L = L ± n (q), 32 for L = S 2n (q), O 2n+1 (q), 38 for L = O + 2n (q) and 36 for O − 2n (q) (cf. the number n 0 given in Tables 1-6).

3.4.
Simple groups for which the recognition problem is not solved. Suppose that L is a simple group for which the recognition problem is not solved, G is a finite group isospectral to L and S is the unique nonabelian composition factor of G. By Theorem 2.1 and the preceding part of this section, L is a classical group in odd characteristic and the open part of recognition problem is quasirecognizability of L. More precisely, either L = S 8 (7 m ) and we refer the reader to the remark after Problem 3.9, or L = S 8 (7 m ) and the question is whether S can be a group of Lie type in characteristic different from the characteristic of L. We conjecture that the answer to this question is negative. If G is a finite group isospectral to L, then the unique nonabelian composition factor S of G cannot be a group of Lie type in characteristic coprime to q.
There are some partial results on Conjecture 3.10: S is not E 8 (u), E 7 (u) by [112] and S is not L 2 (u), S 4 (u), G 2 (u), 2 B 2 (u), 3 D 4 (u) by [46,Proposition 4.8]. Also by Table 7, the recognition problem is solved for some of the groups L in question with q = 3 or 5.
Observe that in the proof of Theorem 2.1 for L = L ± n (q), S 2n (q), O 2n+1 (q), or O ± 2n (q), the condition n n 0 is necessary to prove that S ≃ L but it does not influence the description of almost simple groups with socle L isospectral to L by [33][34][35]. Thus Conjecture 3.10 is in fact is equivalent to Conjecture 3.11. Theorem 2.1 remains valid if we replace n 0 by 5 in Tables 1-6. Conjecture 3.11 covers all nonabelian simple groups except S 8 (7 m ), so the validation of this conjecture and the solution of Problem 3.9 will finalize the study of recognition of simple groups by spectrum.
3.5. Unrecognizable simple groups. The necessary and sufficient condition for a finite group to be unrecognizable by spectrum was established by V. D. Mazurov and W. Shi [85]. Suppose that a simple group L is isospectral to a finite group G with nontrivial solvable radical. Using Theorem 3.12, we can construct infinitely many finite groups isospectral to L but it is not quite clear how to describe all finite groups isospectral to L.
One way is to consider not all the groups isospectral to L but only minimal in some sense. In the same work [85], V. D. Mazurov and W. Shi introduced the following notion: if ω is a finite set of natural numbers, then a finite group G is said to be ω-critical if ω = ω(G) and ω = ω(H) for any proper section H of G (that is, for any H = A/B, where 1 ≤ A B ≤ G and either A = 1, or B = G). Also they proved that for every ω, there are only finitely many ω-critical finite groups and at most one simple ω-critical group. Given an unrecognizable simple group L, we refer to ω(L)-critical groups as just critical. The critical groups are known for Alt 6 [64], Alt 10 [65], J 2 [65], 3 D 4 (2) [76] and L 3 (3) [64]. Some critical groups are known for U 3 (3) [66] and S 4 (3) [68].
We choose to describe groups isospectral to unrecognizable simple groups in the following way. In the first column of Table 10, we list all known unrecognizable simple groups L. It is clear that there are almost simple groups with socle L isospectral to L, and these are known by Theorem 3.7, so in the next three columns we give information about groups G that are isospectral to L but are not almost simple groups with socle L. For solvable groups G, we just indicate whether G is Frobenius or 2-Frobenius. For nonsolvable G, we list all possible combinations of π(K), S and G/H. In the column 'Examples', for every combination, we provide an example of such a group G in notation of [20].
It should be noted that the ristrictions on π(K), S and G/H in the case L = O 9 (2 m ) are known but there are no published proofs of them. Also we note that the list of unrecognizable groups in [78, Table 1], as well as that in [66, Table 1], includes the group L 4 (13 24 ) due to [134]. But, in fact, this group is almost recognizable by [40,46].
Finally, we remark that under Conjecture 3.10, every unrecognizable simple group is either listed in Table 10, or isomorphic to S 8 (7 m ) (cf. Problem 3.9).

Related questions
4.1. Recognition outside the class of finite groups. There are some natural questions related to the problem of recognizing simple groups by spectrum. The first question arises when we enlarge the class of groups in which we seek groups isospectral to a given simple group.
A natural extension of the class of finite groups is the class of locally finite groups. Theorem 3.12 has a simple but useful corollary concerning recognizability of groups in the class of locally finite groups. Proof. (i) Suppose that there is a locally finite but not finite group H isospectral to G. For every a ∈ ω(G), let g a ∈ H be of order a. Then H 0 = g a | a ∈ ω(G) is a finite group isospectral to G.
. . , we obtain infinitely many finite groups H i isospectral to G, contrary to the assumption h(G) < ∞.
(ii) By Theorem 3.12, there is a finite group H isospectral to G and having a nontrivial elementary abelian normal subgroup V . Furthermore, we have an ascending chain of finite groups H i isospectral to G:

The group ∪ ∞
i=1 H i is an infinite locally finite group isospectral to G. By Corollary 4.1, if a finite group G is (almost) recognizable in the class of finite groups then G is (almost) recognizable in the class of locally finite groups. This becomes false for the class of periodic groups. For example, by [82,Theorem 1], there exist infinitely many prime powers q such that L 2 (q) is isospectral to an infinite group (recall from Section 2, that all L 2 (q), except for L 2 (9), are recognizable in the class of finite groups). The deep and difficult problem of recovering a periodic group from its spectrum (it is worth remembering that the finiteness of the Burnside group B(2, 5) is still in question) lies outside the scope of this survey, and we refer the reader to [50,69,84].

4.2.
Recognition of almost simple groups. The second question is what non-simple finite groups are recognizable or almost recognizable by spectrum. Recall that the socle Soc(G) of a finite group G is the subgroup generated by all minimal normal subgroups of G. By Theorem 3.12, if G is an almost recognizable finite group, then for some nonabelian simple groups L i , 1 i k. Assume first that k = 1 in (4.1), that is, G is an almost simple group. The recognition problem is solved for all symmetric groups Sym n except Sym 10 (see [25,27] and references therein). Namely, they are unrecognizable if n ∈ {2, 3, 4, 5, 6, 8} and recognizable otherwise. By [81,87], if L is a sporadic group and L = J 2 , then Aut L is recognizable. Observe that h(Aut J 2 ) is either 1 or ∞ but the exact value of this number is still not known (see [139] for more details). We now turn to almost simple groups of Lie type. Clearly, numerous examples of almost recognizable almost simple groups of this sort are given by Theorem 2.1, namely, these are non-simple almost simple groups isospectral to simple groups L of Item (ii) of this theorem. On the other hand, applying known facts on spectra of almost simple groups, one can easily find infinitely many unrecognizable almost simple groups. For example, let q be a power of a prime p, n an odd number such that n − 1 is not a power of p and G = P GL ε n (q). Then ω(G) = ω(SL ε n (q)) [14, Proposition 2.6]. If, in addition, (n, q − ε1) = 1, then the center of SL ε n (q) is nontrivial, so the group G is unrecognizable [14,Corollary 4]. One of the natural obstacles to solving the recognition problem for almost simple groups is that despite some progress (see, for example, [36]), the following problem is still open. Even when Soc(G) = L 2 (q), the only general result is that h(P GL 2 (q)) = ∞ if q is a prime or q = 9, and h(P GL 2 (q)) = 1 otherwise [19]. Also the problem is solved for q = 4, 5, in which case G = Sym 5 , and for q = 9, in which case G is one of the groups Sym 6 , P GL 2 (9), M 10 , and Aut(L 2 (9)). By the above, h(Sym 6 ) = h(P GL 2 (9)) = ∞. By [11], we have h(M 10 ) = 1. Since ω(Aut(L 2 (9))) = ω(SL 2 (9)), it follows that h(Aut(L 2 (9))) = ∞.

4.3.
Recognition of groups with non-simple socle. Let now G be a group satisfying (4.1) with k > 1. Generally, one can hardly expect the group G to be recognizable or almost recognizable. For example, let G i = Alt 5 × Alt 4+i with i = 1, 2. Clearly, µ(G i ) = {2 i · 3, 2 i · 5, 3 · 5}, where µ(G) is the subset of ω(G) consisting of numbers maximal under divisibility. So G i is unrecognizable being isospectral to the direct product of the Frobenius groups of the form 5 2 : 2 i and 5 2 : 3 (their kernels are elementary abelian groups of order 25 and complements are cyclic group of orders 2 i and 3 respectively). Nevertheless, if we do not presuppose that the simple factors of the socle of G are isomorphic to each other, then we can construct a recognizable group with k arbitrarily large.
Define primes p i and groups L i in the following way.
(4.2) Let p 1 = 7 and L 1 = 2 B 2 (2 p 1 ). For i > 1, define p i to be the smallest prime larger than p i−1 and not lying in ∪ j<i π(L j ), and put L i = 2 B 2 (2 p i ) The order, the spectrum and subgroups of the simple Suzuki groups are well-known and can be found in [113]. We have | 2 B 2 (q)| = q(q − 1)(q 2 + 1), and so π(L i ) ∩ π(L j ) = {2, 5} if i = j. Also p i ∈ π(L j ) for all i and j. Since µ( 2 B 2 (q)) = {4, q − 1, q − √ 2q + 1, q + √ 2q + 1}, if we take r i1 ∈ π(q i − 1), r i2 ∈ π(q i − √ 2q i + 1) and r i3 ∈ π(q i + √ 2q i + 1), where q i = 2 p i , then {r i1 , r i2 , r i3 } is a coclique in GK(L i ). Observe that we can take r i2 and r i3 to be larger than 5 because p i 7. Now let G k = k i=1 2 B 2 (q i ) and let G be a finite group isospectral to G k . It is clear that {r i1 , r i2 , r i3 } is still a coclique in GK(G k ), so t(G) 3, and hence G is not solvable by [51]. Let K be the solvable radical of G and let S 1 × · · · × S t be the socle of G/K. Since 3 ∈ π(G k ), we see that every S i is equal to 2 B 2 (2 m i ) for some m i . By the Bang-Zsigmondy theorem [10], for every m 7, there is a prime dividing 2 m − 1 but not 2 l − 1 for l < m. Using this fact, it is not hard to see that {m 1 , . . . , m t } ⊆ {p 1 , . . . , p k }. If m i = m j = p l for some i = j, then r l1 r l2 ∈ ω(G) \ ω(G k ), so m i = m j if i = j. In particular, G/K Aut S 1 × · · · × Aut S t . Observe that | Out L i | = p i , so G/K = Soc(G/K). If some p i ∈ {m 1 , . . . , m t }, then {r i1 , r i2 , r i3 } ⊆ π(K) and t(K) 3, which is a contradiction since K is solvable. Thus G/K is isomorphic to G k , and we may assume that S i ≃ L i for all 1 i k.
Suppose that K = 1. There are a prime r and a normal subgroup K 0 of K such that V = K/K 0 is an elementary abelian r-group. For every 1 i k, let x i be an element of G of order r i2 if r not dividing q i − √ 2q i + 1, or of order r i3 otherwise. Let V 0 = V and V 1 = C V (x 1 ). Then S 2 acts on K 1 and we define V 2 = C V 1 (x 2 ) and so on. If V k = 1, then G contains an element of prohibited order r|x 1 | . . . |x k |. Suppose that V k = 1 and let i be the smallest i such that V i = 1. It is clear that the image of x i in G/K lies in S i , so by [46,Lemma 3.8], it follows that r = 2. The group S i includes a Frobenius subgroup with kernel x i and cyclic complement of order 4. Since C V i−1 (x i ) = 1, by [70,Lemma 1], we conclude that G has an element of order 8, which is a contradiction. Thus K = 1, and so we prove the following Theorem 4.6. For every k 1, the group k i=1 2 B 2 (2 p i ), where p i are the primes as in (4.2), is recognizable by spectrum.
Let now L = L 1 ≃ . . . ≃ L k and so Soc(G) = L k . We begin with the case when G = Soc(G), that is, G is a product of nonabelian simple groups. The following theorem collects all known examples of recognizable groups of this type.
Theorem 4.7. The following groups are recognizable by spectrum: Note that for every finite simple group L, there exists k (depending on L) such that G = L k is unrecognizable. Indeed, if k ≥ |µ(L)|, then µ(G) = {m} is a one-element set and so H is unrecognizable being isospectral to a cyclic group of order m. For example, µ(Alt 5 ) = {2, 3, 5}, so (Alt 5 ) 3 is isospectral to a cyclic group of order 30 (as we remarked above, (Alt 5 ) 2 is also unrecognizable). The question about existence of a global constant c such that L c is unrecognizable for all finite simple groups L is open. We put it as follows.
Problem 4.8. Does there exist a finite recognizable by spectrum group which is a direct product of k copies of a finite nonabelian simple group for arbitrarily large k?
The case G = Soc(G) is even more mysterious. In [102] W. Shi conjectured that the group G having a minimal normal subgroup N with |µ(N)| = 1 must be unrecognizable. However, as shown by V. D. Mazurov in [71], if G is the permutation wreath product of the simple Suzuki group B = 2 B 2 (2 7 ) and a subgroup of the symmetric group on 23 letters which is isomorphic to a Frobenius group of order 23 · 11, then h(G) = 1. This gives a counterexample to Shi's conjecture because |µ(B)| = 4, and so the minimal normal subgroup N = Soc(G) = B 23 of G has |µ(N)| = 1. It would be interesting to generalize Mazurov's example replacing a subgroup of Sym 23 with a subgroup of Sym k , where k belongs to some infinite series, possibly with another group B. Problem 4.9. Does there exist a finite simple group B such that for infinitely many numbers k, there is a finite recognizable by spectrum group G with Soc(G) = B k ?
We would like to emphasize the difference between Problem 4.8 and Problem 4.9: in the former, a new simple group can be chosen for every new k, while in the latter, we need one fixed simple group suitable for infinitely many k.
Observe that for all recognizable groups G mentioned above, the group G/Soc(G) is solvable. We conclude our survey with the following question posed by D. O. Revin.