Finite Quasisimple Groups Research Articles

EXTENSIONS OF CHARACTERS IN TYPE D AND THE INDUCTIVE MCKAY CONDITION, I

AbstractThis is a contribution to the study of$\mathrm {Irr}(G)$as an$\mathrm {Aut}(G)$-set forGa finite quasisimple group. Focusing on the last open case of groups of Lie type$\mathrm {D}$and$^2\mathrm {D}$, a crucial property is the so-called$A'(\infty )$condition expressing that diagonal automorphisms and graph-field automorphisms ofGhave transversal orbits in$\mathrm {Irr}(G)$. This is part of the stronger$A(\infty )$condition introduced in the context of the reduction of the McKay conjecture to a question about quasisimple groups. Our main theorem is that a minimal counterexample to condition$A(\infty )$for groups of type$\mathrm {D}$would still satisfy$A'(\infty )$. This will be used in a second paper to fully establish$A(\infty )$for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of any standard Levi subgroup of$G=\mathrm {D}_{ l,\mathrm {sc}}(q)$extend to their stabilizers in the normalizer of that Levi subgroup. This allows us to control the action of automorphisms on these extensions. From there, Harish-Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context.

Unitriangular basic sets, Brauer characters and coprime actions

We show that the decomposition matrix of a given groupGGis unitriangular, wheneverGGhas a normal subgroupNNsuch that the decomposition matrix ofNNis unitriangular,G/NG/Nis abelian and certain characters ofNNextend to their stabilizer inGG. Using the recent result by Brunat–Dudas–Taylor establishing that unipotent blocks have a unitriangular decomposition matrix, this allows us to prove that blocks of groups of quasi-simple groups of Lie type have a unitriangular decomposition matrix whenever they are related via Bonnafé–Dat–Rouquier’s equivalence to a unipotent block. This is then applied to study the action of automorphisms on Brauer characters of finite quasi-simple groups. We use it to verify the so-calledinductive Brauer–Glauberman conditionthat aims to establish a Glauberman correspondence for Brauer characters, given a coprime action.

The essential p-dimension of the split finite quasi-simple groups of classical Lie type

In this paper, we compute the essential p-dimension of the split finite quasi-simple groups of classical Lie type at the defining prime, specifically the quasi-simple groups arising from the general linear and special linear groups, the symplectic groups, and the orthogonal groups.

Finite quasisimple groups acting on rationally connected threefolds

AbstractWe show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: ${\mathfrak{A}}_5$ , ${\text{PSL}}_2(\textbf{F}_7)$ , ${\mathfrak{A}}_6$ , ${\text{SL}}_2(\textbf{F}_8)$ , ${\mathfrak{A}}_7$ , ${\text{PSp}}_4(\textbf{F}_3)$ , ${\text{SL}}_2(\textbf{F}_{7})$ , $2.{\mathfrak{A}}_5$ , $2.{\mathfrak{A}}_6$ , $3.{\mathfrak{A}}_6$ or $6.{\mathfrak{A}}_6$ . All of these groups with a possible exception of $2.{\mathfrak{A}}_6$ and $6.{\mathfrak{A}}_6$ indeed act on some rationally connected threefolds.

Commuting involutions and elementary abelian subgroups of simple groups

Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that contains a member of each conjugacy class of involutions of G.

A note on the normal subgroup lattice of ultraproducts of finite quasisimple groups

Abel Stolz and Andreas Thom [Proc. Lond. Math. Soc. 108 (2014), pp. 73–102] stated that the lattice of normal subgroups of an ultraproduct of finite simple groups is always linearly ordered. This is false in this form in most cases for classical groups of Lie type. We correct the statement in this case and point out a version of \enquote*relative bounded normal generation for classical quasisimple groups and its implications on the structure of the lattice of normal subgroups of an ultraproduct of such groups.

On Willems' conjecture on Brauer character degrees

In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for the sum of squares of the degrees of the irreducible p-Brauer characters of a finite group G. We prove this conjecture for the prime p=2. For this we rely on the recent reduction of Willems' conjecture to a question on quasi-simple groups by Tong-Viet. We also verify the conditions of Tong-Viet for certain families of finite quasi-simple groups and odd primes. On the way we obtain lower bounds for the number of regular semisimple conjugacy classes in finite groups of Lie type.

On maximal embeddings of finite quasisimple groups

If a finite quasisimple group G with simple quotient S is embedded into a suitable classical group X through the smallest degree of a projective representation of S, then is a maximal subgroup of X, up to two series of exceptions where S is a Ree group, and four exceptions where S is sporadic.

Isomorphism questions for metric ultraproducts of finite quasisimple groups

Abstract New results on metric ultraproducts of finite simple groups are established. We show that the isomorphism type of a simple metric ultraproduct of groups X n i ⁢ ( q ) {X_{n_{i}}(q)} ( i ∈ I {i\in I} ) for X ∈ { PGL , PSp , PGO ( ε ) , PGU } {X\in\{\operatorname{PGL},\operatorname{PSp},\operatorname{PGO}^{(\varepsilon)% },\operatorname{PGU}\}} ( ε = ± {\varepsilon=\pm} ) along an ultrafilter 𝒰 {\mathcal{U}} on the index set I for which n i → 𝒰 ∞ {n_{i}\to_{\mathcal{U}}\infty} determines the type X and the field size q up to the possible isomorphism of a metric ultraproduct of groups PSp n i ⁡ ( q ) {\operatorname{PSp}_{n_{i}}(q)} and a metric ultraproduct of groups PGO n i ( ε ) ⁡ ( q ) {\operatorname{PGO}_{n_{i}}^{(\varepsilon)}(q)} . This extends results of [A. Thom and J. Wilson, Metric ultraproducts of finite simple groups, Comp. Rend. Math. 352 2014, 6, 463–466].

Almost cyclic elements in cross-characteristic representations of finite groups of Lie type

Abstract This paper is a significant contribution to a general programme aimed to classify all projective irreducible representations of finite simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix (that is, a square matrix M of size n over a field 𝔽 {\mathbb{F}} with the property that there exists α ∈ 𝔽 {\alpha\in\mathbb{F}} such that M is similar to diag ⁡ ( α ⋅ Id k , M 1 ) {\operatorname{diag}(\alpha\cdot\mathrm{Id}_{k},M_{1})} , where M 1 {M_{1}} is cyclic and 0 ≤ k ≤ n {0\leq k\leq n} ). While a previous paper dealt with the Weil representations of finite classical groups, which play a key role in the general picture, the present paper provides a conclusive answer for all cross-characteristic projective irreducible representations of the finite quasi-simple groups of Lie type and their automorphism groups.

On Bogomolov multiplier and Ш-rigidity of groups

We show the existence of groups with trivial Bogomolov multiplier such that the Bogomolov multiplier of a central product of these groups is non-trivial. We then prove that finite quasisimple groups and freest special p-groups are Ш-rigid in the sense that all their class-preserving automorphisms are inner. As an application we show the existence of non-nilpotent Ш-rigid groups and Ш-rigid p-groups of odd order having non-trivial Bogomolov multiplier.

Towards the Eaton-Moretó conjecture on the minimal height of characters

In this note we prove that the Eaton-Moretó conjecture holds for all blocks of finite general linear and unitary groups for all primes. Also, we show that no block of a finite quasi-simple group of classical Lie type provides a minimal counterexample to the conjecture, and so for ℓ > 5 no ℓ-block of any quasi-simple group can be a minimal counterexample.

On the Number of Characters in Blocks of Quasi-simple Groups

We prove, for primes p ≥ 5, two inequalities between the fundamental invariants of Brauer p-blocks of finite quasi-simple groups: the number of characters in the block, the number of modular characters, the number of height 0 characters, and the number of conjugacy classes of a defect group and of its derived subgroup. For this, we determine these invariants explicitly, or at least give bounds for them for several classes of classical groups.

On a minimal counterexample to Brauer’s k(B)-conjecture

We study Brauer’s long-standing k(B)-conjecture on the number of characters in p-blocks for finite quasi-simple groups and show that their blocks do not occur as a minimal counterexample for p ≥ 5 nor in the case of abelian defect. For p = 3 we obtain that the principal 3-blocks do not provide minimal counterexamples. We also determine the precise number of irreducible characters in unipotent blocks of classical groups for odd primes.

Imprimitive irreducible modules for finite quasisimple groups. II

This work completes the classification of the imprimitive irreducible modules, over algebraically closed fields of characteristic 00, of the finite quasisimple groups.

The length and depth of algebraic groups

Let G be a connected algebraic group. An unrefinable chain of G is a chain of subgroups G = G_0> G_1> cdots > G_t = 1, where each G_i is a maximal connected subgroup of G_{i-1}. We introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group G in terms of the dimension of its unipotent radical R_u(G) and the dimension of a Borel subgroup B of the reductive quotient G/R_u(G). In particular, a simple algebraic group of rank r has length dim B + r, which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group G exceeds frac{1}{2} dim G. We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most 6 (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group G with soluble radical R(G), the dimension of G / R(G) is bounded above in terms of the chain difference of G.

Varieties of elements of given order in simple algebraic groups

Given a positive integer u and a simple algebraic group G defined over an algebraically closed field K of characteristic p, we derive properties about the subvariety G[u] of G consisting of elements of G of order dividing u. In particular, we determine the dimension of G[u], completing results of Lawther [7] in the special case where G is of adjoint type. We also apply our results to the study of finite simple quotients of triangle groups, giving further insight on a conjecture we proposed in [10] as well as proving that some finite quasisimple groups are not quotients of certain triangle groups.

Surjective word maps and Burnside’s $$p^aq^b$$ p a q b theorem

We prove surjectivity of certain word maps on finite non-abelian simple groups. More precisely, we prove the following: if N is a product of two prime powers, then the word map $$(x,y) \mapsto x^Ny^N$$ is surjective on every finite non-abelian simple group; if N is an odd integer, then the word map $$(x,y,z) \mapsto x^Ny^Nz^N$$ is surjective on every finite quasisimple group. These generalize classical theorems of Burnside and Feit–Thompson. We also prove asymptotic results about the surjectivity of the word map $$(x,y) \mapsto x^Ny^N$$ that depend on the number of prime factors of the integer N.

Almost cyclic elements in Weil representations of finite classical groups

ABSTRACTThis paper is a significant part of a general project aimed to classify all irreducible representations of finite quasi-simple groups over an algebraically closed field, in which the image of at least one element is represented by an almost cyclic matrix (that is, a square matrix M of size n over a field F with the property that there exists α∈F such that M is similar to , where M1 is cyclic and 0≤k≤n). The paper focuses on the Weil representations of finite classical groups, as there is strong evidence that these representations play a key role in the general picture.

Quasi-simple groups all of whose non-principal blocks are of defect zero

Abstract In this paper we characterize finite quasi-simple groups of p-deficiency class 1 (i.e., quasi-simple groups all of whose non-principal p-blocks are of defect zero) for an odd prime p.