The maximal size of a minimal generating set

Rank one isolated p-minimal subgroups in finite groups

Let G be a finite group and cd(G) be the set of all irreducible complex character degrees of G without multiplicities. The aim of this paper is to propose an extension of Huppert’s conjecture from non-Abelian simple groups to almost simple groups of Lie type. Indeed, we conjecture that if H is an almost simple group of Lie type with cd(G)=cd(H), then there exists an Abelian normal subgroup A of G such that G/A≅H. It is furthermore shown that G is not necessarily the direct product of H and A. In view of Huppert’s conjecture, we also show that the converse implication does not necessarily hold for almost simple groups. Finally, in support of this conjecture, we will confirm it for projective general linear and unitary groups of dimension 3.

In this paper we prove that if G is a finite exceptional simple group of Lie type, then G admits a 2-covering if, and only if, it is one of the following groups: \({G_2(2^{a}), F_4(3^{a}), G_2(2)', ^{2}G_2(3)', ^{2}F_4(2)'}\). Furthermore, if G is a finite sporadic simple group, then G admits a 2-covering if, and only if, G = M11.

Let G be a simple group of Lie type over an infinite locally finite field F. For any field K, we prove that the group algebra K [ G ] K[G] is semiprimitive. The argument here is a mixture of combinatorial and topological methods. Combined with earlier results, it now follows that any group algebra of an infinite locally finite simple group is semiprimitive. Furthermore, if the group is countably infinite, then the group algebra is primitive. In particular, if G is a simple group of Lie type over the field F, then K [ G ] K[G] is a primitive ring.

We provide new bounds for the divisibility function of the free group $${\mathbf F}_2$$ and construct short laws for the symmetric groups $${{\mathrm{Sym}}}(n)$$ . The construction is random and relies on the classification of the finite simple groups. We also give bounds on the length of laws for finite simple groups of Lie type.

We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from mathbf {PSL}_n(q) collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is mathbf {PSp}_{2n}(q), mathbf {P}{varvec{Omega }}^+_{4n}(q), mathbf {P}{varvec{Omega }}^-_{4n}(q), ^3D_4(q), E_7(q), E_8(q), F_4(q), or G_2(q) with q even is the group algebra.

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 $$I_n(G)$$ 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.

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.

We study the subgroup B 0 ( G ) of H 2 ( G , [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]) consisting of all elements which have trivial restrictions to every Abelian subgroup of G . The group B 0 ( G ) serves as the simplest nontrivial obstruction to stable rationality of algebraic varieties V/G where V is a faithful complex linear representation of the group G . We prove that B 0 ( G ) is trivial for finite simple groups of Lie type A ℓ .

We prove that all finite simple groups of Lie type, with the exception of the Suzuki groups, can be made into a family of expanders in a uniform way. This confirms a conjecture of Babai, Kantor and Lubotzky from 1989, which has already been proved by Kassabov for sufficiently large rank. The bounded rank case is deduced here from a uniform result for SL2 which is obtained by combining results of Selberg and Drinfeld via an explicit construction of Ramanujan graphs by Lubotzky, Samuels and Vishne.

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for families of simple groups of Lie type of bounded rank, we obtain positive results. For example, for any fixed untwisted Lie type X there is an algorithm that determines whether or not any given finitely presented group has simple images of the form X(q) for infinitely many q, and if there are finitely many, the algorithm determines them.

The degree pattern of a finite group G is introduced in [10] and it is proved that the following simple groups are uniquely determined by their degree patterns and orders: all sporadic simple groups, alternating groups Ap (p ≥ 5 is a twin prime) and some simple groups of Lie type. In this paper, we continue this investigation. In particular, we show that the automorphism groups of sporadic simple groups (except Aut (J2) and Aut (McL)), all simple C22-groups, the alternating groups Ap, Ap+1, Ap+2 and the symmetric groups Sp, Sp+1, where p is a prime, are also uniquely determined by their degree patterns and orders.

In this paper we describe the connected components of Nl(G) , the partially ordered set of nilpotent subgroups of a finite simple group of Lie type.Key words: prime graphs, simplicial

Finite simple nonabelian groups G that are not π-closed for some set of primes π but have π-closed maximal subgroups (property (*) for (G, π)) are studied. We give a list L of finite simple groups that contains any group G with the above property (for some π). It is proved that 2 ∉ π for any pair (G, π) with property (*) (Theorem 1). In addition, we specify for any sporadic simple group G from L all sets of primes π such that the pair (G, π) has property (*) (Theorem 2). The proof uses the author’s results on the control of prime spectra of finite simple groups.