Characteristically simple ℵ0-categorical groups

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, Professor J. G. Thompson gave the following conjecture. Thompson's Conjecture. If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G):={n ∈ ℕ | G has a conjugacy class of size n}. In 2006, A. Abdollahi, S. Akbari, and H. R. Maimani put forward a conjecture (AAM's conjecture) in Abdollahi et al. (2006) as follows. AAM's Conjecture. Let M be a finite nonabelian simple group and G a group such that ∇(G) ≅ ∇ (M). Then G ≅ M. In this short article we prove that if G is a finite group with ∇(G) ≅ ∇ (A 10), then G ≅ A 10, where A 10 is the alternating group of degree 10.

We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.

We survey recent results regarding embeddings of finite simple groups (and their nonsplit central extensions) in complex Lie groups, especially the Lie groups of exceptional type. Introduction Throughout this paper, L will be a finite group. Representation theory for L is usually understood to be the study of group morphisms L → GL(n, k) for distinguished collections of fields k (e.g., all overfields of a fixed field F ) and positive integers n . The topic of this survey is motivated by the question as to what happens if GL ( n ,·) is replaced by another algebraic group G (·). We shall mainly be concerned with the case where L is a finite simple group (that is, a finite nonabelian simple group) or a central extension thereof, and G(k) is a connected simple algebraic group over a field k . A further restriction of our discussion concerns the field k . It will mostly be taken to be the complex numbers, in which case we will mainly study group morphisms from L to the complex Lie group G (ℂ). (See below for some exceptions in §3 and §5.) For G (·) of classical type, the theory for representations L → G (ℂ) differs little from the usual one for GL ( n , ℂ). Indeed, a representation L → GL ( n , ℂ) decomposes into irreducible subrepresentations. The decomposition is well controlled by character theory.

Let G be a finite nonabelian group and associate a disoriented noncommuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In 1987, J. G. Thompson gave the following conjecture.Thompson's Conjecture If G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying N(G) = N(M), then G ≅ M, where N(G) denotes the set of the sizes of the conjugacy classes of G.In 2006, A. Abdollahi, S. Akbari and H. R. Maimani put forward a conjecture in [1] as follows.AAM's Conjecture Let M be a finite nonabelian simple group and G a group such that ∇(G)≅ ∇ (M). Then G ≅ M.Even though both of the two conjectures are known to be true for all finite simple groups with nonconnected prime graphs, it is still unknown for almost all simple groups with connected prime graphs. In the present paper, we prove that the second conjecture is true for the projective special unitary simple group U4(7).

The set π(G) of all prime divisors of the order of a finite group G is often called its prime spectrum. It is proved that every finite simple nonabelian group G has sections H1, …, Hm of some special form such that π(H1)∪…∪π(Hm) = π(G) and m ≤ 5. Moreover, m ≤ 2 if G is an alternating or classical simple group. In all cases, it is possible to choose the sections Hi so that each of them is a simple nonabelian group, a Frobenius group, or (in one case) a dihedral group. If the above equality holds for a finite group G, then we say that the set {H1,…,Hm} controls the prime spectrum of G. We also study some parameter c(G) of finite groups G related to the notion of control.

In her book on varieties of groups, Hanna Neumann posed the following problem [13, p. 166]: “Can a variety other than D contain an infinite number of non-isomorphic non-abelian finite simple groups?”The answer to this question does not seem to be known at present. However, in [7], Heineken and Neumann described an algorithm for determining whether or not there are any non-abelian finite simple groups satisfying a given law. They also outlined a way in which their algorithm could be used to show that “only finitely many of the known non-abelian finite simple groups can satisfy a given non-trivial law”; in this paper, we shall follow their suggestions, and prove theTHEOREM. Let g be a set of mutually non-isomorphic non-abelian finite simple groups, each of which is either an alternating group or a group of Lie type, and let g generate a proper subvariety of D. Then y is finite.

In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.

In this paper we introduce the conjugate graph [Formula: see text] associated to a nonabelian group G with vertex set G\Z(G) such that two distinct vertices join by an edge if they are conjugate. We show if [Formula: see text], where S is a finite nonabelian simple group which satisfy Thompson's conjecture, then G ≅ S. Further, if central factors of two nonabelian groups H and G are isomorphic and |Z(G)| = |Z(H)|, then H and G have isomorphic conjugate graphs.

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Abdollahi et al. (J Algebra 298(2):468–492, 2006) put forward a conjecture called AAM’s Conjecture in as follows: If M is a finite nonabelian simple group and G is a group such that ∇(G) ≅ ∇(M), then G ≅ M. Even though this conjecture is well known to hold for all simple groups with nonconnected prime graphs and the alternating group A10 [see Darafsheh (Groups with the same non-commuting graph. Discrete Appl Math (2008) doi:10.1016/j.dam.2008.06.010), Wang and Shi (Commun Algebra 36(2):523–528, 2008)], it is still unknown for all simple groups with connected prime graphs except A10. In the present paper, we prove that this conjecture is also true for the projective special linear simple group L4(9). The new method used in this paper also works well in the cases L4(4), L4(7), U4(7), etc.

Throughout this note, G will be a finite group, IrrðGÞ will be the set of irreducible characters of G, cdðGÞ will be the set of character degrees of G, and rðGÞ will be the set of primes that divide degrees in cdðGÞ. The prime vertex degree graph of G, written DðGÞ, is the graph with rðGÞ as its vertex set, and with an edge between p and q if pq divides some degree a A cdðGÞ. An overview of the literature on these graphs can be found in [2]. It seems that most groups G have graphs DðGÞ that are complete graphs (although we do not want to be precise on what we mean by ‘most’). For solvable groups, it has been shown that if DðGÞ is not complete, then the structure of G is limited (see [4] and [6]). In this note, we show that a large class of non-solvable groups have degree graphs that are complete. To study the degree graphs of non-solvable groups, a standard starting point is with the degree graphs of non-abelian simple groups. In a series of papers, Don White has classified the character degree graphs of all non-abelian simple groups. Except for some examples of small rank, nearly all non-abelian simple groups have degree graphs that are complete graphs. These results are summarized in [5]. Extending this idea, this note looks at the characteristically simple groups, groups for which there are exactly two characteristic subgroups. Each such group is a direct product of copies of a fixed simple group. With this in mind, we let G denote G G (n copies); thus S is characteristically simple when S itself is a simple group. We now fix S to be a non-abelian simple group. Because the non-abelian simple groups have already been addressed, we restrict our attention to when n > 1. However, it is relatively easy to show that DðSÞ is complete for n > 1. More generally, the graph DðHÞ associated with any non-abelian group H is always complete for every integer n > 1. Thus we expand our scope and consider extending these groups. We recognize first that S is isomorphic to InnðSÞ. Identifying S with its set of inner automorphisms, our goal herein is to prove the following result.

Let $$I_n(G)$$ denote the number of elements of order n in a finite group G. In 1979, Herzog (Proc Am Math Soc 77:313–314, 1979) conjectured that two finite simple groups containing the same number of involutions have the same order. In a 2018 paper (Arch Math 111:349–351, 2018), Zarrin disproved Herzog’s conjecture with a counterexample. Then he conjectured that “if S is a non-abelian simple group and G a group such that $$I_2(G)=I_2(S)$$ and $$I_p(G) =I_p(S)$$ for some odd prime divisor p, then $$|G|=|S|$$ ”. In this paper, we give more counterexamples to Herzog’s conjecture. Moreover, we disprove Zarrin’s conjecture.

For a finite group G, it is denoted by N(G) the set of conjugacy class sizes of G. In 1980s, J. G. Thompson posed the following conjecture: if L is a finite nonabelian simple group, G is a finite group with trivial center, and N(G) = N(L), then L and G are isomorphic. In this paper, it is proved that Thompson’s conjecture is true for the alternating group A22 with connected prime graph.

Finite groups which possess a strongly closed 2-subgroup of class at most two, I

Simple classical groups of Lie type are determined by their character degrees

The noncommuting graph ∇(G) of a non-abelian group G is defined as follows. The vertex set of ∇(G) is G\Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy ≠ yx. It has been conjectured that if P is a finite non-abelian simple group and G is a group such that ∇(P) ≅ ∇(G), then G ≅ P. In the present paper, our aim is to consider this conjecture in the case of finite almost simple groups. In fact, we characterize the projective general linear group PGL (2, q) (q is a prime power), which is also an almost simple group, by its noncommuting graph.