Finite simple groups of Lie type as expanders

A generating set for a finite group G is minimal if no proper subset generates G, and $m(G)$ denotes the maximal size of a minimal generating set for G. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist $a,b> 0$ such that any finite group G satisfies $m(G) \leqslant a \cdot \delta (G)^b$ , for $\delta (G) = \sum _{p \text { prime}} m(G_p)$ , where $G_p$ is a Sylow p-subgroup of G. To do this, we first bound $m(G)$ for all almost simple groups of Lie type (until now, no nontrivial bounds were known except for groups of rank $1$ or $2$ ). In particular, we prove that there exist $a,b> 0$ such that any finite simple group G of Lie type of rank r over the field $\mathbb {F}_{p^f}$ satisfies $r + \omega (f) \leqslant m(G) \leqslant a(r + \omega (f))^b$ , where $\omega (f)$ denotes the number of distinct prime divisors of f. In the process, we confirm a conjecture of Gill and Liebeck that there exist $a,b> 0$ such that a minimal base for a faithful primitive action of an almost simple group of Lie type of rank r over $\mathbb {F}_{p^f}$ has size at most $ar^b + \omega (f)$ .

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.

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 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 .

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

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.

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.

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 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.

Let X ( r ) X(r) and G ( q ) G(q) be finite groups of Lie type and r r and q q be coprime. If G ( q ) G(q) is embedded in X ( r ) X(r) , then the Landazuri-Seitz-Zalesskii theorem implies that G ( q ) G(q) is small relative to X ( r ) X(r) . We formalize this observation and illustrate how it can be used with some applications.

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.

Lie-type-like groups

Prime Graph Components of the Simple Groups of Lie Type over the Field of Even Characteristic

Let 𝐾 be a finite simple group of Lie type over a field of even order q > 2 q>2 . If 𝐾 is not F 4 2 ⁢ ( q ) {}^{2}F_{4}(q) , then we determine the fusion systems ℱ of J-component type with a fully centralized involution 𝑗 such that C F ⁢ ( j ) C_{\mathcal{F}}(j) has a component realized by 𝐾. The exceptional case is treated in a later paper.