Quasisimple Groups Research Articles

On localizations of quasi-simple groups with given countable center

A group homomorphism i\colon H \to G is a localization of H , if for every homomorphism \varphi\colon H\to G there exists a unique endomorphism \psi\colon G\to G such that i \psi=\varphi (maps are acting on the right). Göbel and Trlifaj asked in [18, Problem 30.4(4), p. 831] which abelian groups are centers of localizations of simple groups. Approaching this question we show that every countable abelian group is indeed the center of some localization of a quasi-simple group, i.e., a central extension of a simple group. The proof uses Obraztsov and Ol'shanskii's construction of infinite simple groups with a special subgroup lattice and also extensions of results on localizations of finite simple groups by the second author and Scherer, Thévenaz and Viruel.

RATIONAL NEARLY SIMPLE GROUPS

AbstractA finite group whose irreducible complex characters are rational-valued is called a rational group. The aim of this paper is to determine the rational almost simple and rational quasi-simple groups.

On a Divisibility Property Involving the Sum of Element Orders

A finite group G is called $$\psi $$ -divisible if $$\psi (H)|\psi (G)$$ for any subgroup H of G, where $$\psi (H)$$ and $$\psi (G)$$ are the sum of element orders of H and G, respectively. In this paper, we classify the finite groups whose subgroups are all $$\psi $$ -divisible. Since the existence of $$\psi $$ -divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the $$\psi $$ -divisibility property of ZM-groups. In the end, we introduce the concept of $$\psi $$ -normal divisible group, i.e., a group for which the $$\psi $$ -divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many $$\psi $$ -normal divisible groups which are neither simple nor nilpotent.

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.

ZJ-controlled blocks of quasisimple groups for odd primes

We investigated the converses of Glauberman's ZJ-theorem and its block theoretic analogue, proved by Kessar, Linckelmann and Robinson for a finite group G. We proved that a block B of G with a defect group D is ZJ-controlled (that is, NG(Z(J(D))) controls B-fusion in G) if and only if B is controlled provided that G is quasisimple and the characteristic p of the modular representation is odd. This implies by a result of [5] that the converses of Glauberman's ZJ-theorem and its block theoretic analogue both hold in these cases. As an application we obtain all Qd(p)-free quasisimple groups.

Controlled 2-blocks of quasisimple groups

Controlled p-blocks, introduced by Alperin and Broué in 1979, were classified for quasisimple groups G for odd primes in [10]. We classify the controlled 2-blocks of quasisimple groups G. The results imply that every nilpotent 2-block of G has abelian defect groups. This agrees with one of the main results proved in [15]. We also give an explicit characterization of non-controlled 2-blocks of all quasisimple groups. This implies that the p=2 version of the block theoretic analogue of Glauberman's ZJ-theorem, proved by Kessar, Linckelmann and Robinson in [27], holds for quasisimple groups G.

On heights of characters of finite groups

In this paper we first state a conjecture on the lower bound of the maximal height of characters in a p-block of a finite group. Then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group, for all blocks of a quasi-simple group G with G/Z(G) isomorphic to A6,A7 or a simple group of Lie type with an exceptional covering group, for all blocks of a symmetric group, as well as for all blocks of finite general linear or unitary groups. For the proof of the case with symmetric groups, it relates to an open question of Olsson on the existence of t-core partitions, where t≥3 is an integer. As a byproduct, our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the Isaacs-Moretó-Navarro-Tiep Conjecture, claiming that the number of distinct irreducible character degrees of a Sylow p-subgroup P of an arbitrary finite group G is at most one more than the number of irreducible character degrees of G that are multiples of p.

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].

On the Alperin-McKay conjecture for simple groups of type A

In this paper characters of the normalizer of d-split Levi subgroups in SLn(q) and SUn(q) are parametrized with a particular focus on the Clifford theory between the Levi subgroup and its normalizer. These results are applied to verify the Alperin-McKay conjecture for primes ℓ with ℓ∤6(q2−1) and the Alperin weight conjecture for ℓ-blocks of those quasi-simple groups with abelian defect. The inductive Alperin-McKay condition and inductive Alperin weight condition by the second author are verified for certain blocks of SLn(q) and SUn(q).

Some conjectures on Brauer character degrees

Let G be a finite group and let p be a prime. Willems conjectured that the sum of the squares of all Brauer character degrees of a finite group G is bounded below by the p′-part of the order of G. In this note, we propose a ‘projective’ version of this conjecture and prove it for p-solvable groups. Using this, we manage to reduce the proof of Willems conjecture to a question concerning quasi-simple groups. We also introduce a projective version of a conjecture due to Martínez-Pérez and Willems on projective indecomposable characters which, if true, implies the veracity of the projective version of the Willems conjecture above.

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.

Donovan\u2019s conjecture, blocks with abelian defect groups and discrete valuation rings

We give a reduction to quasisimple groups for Donovan’s conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring {mathcal {O}}. Consequences are that Donovan’s conjecture holds for {mathcal {O}}-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovan’s conjecture for {mathcal {O}}-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect. A result of independent interest is that in general (i.e. for arbitrary defect groups) Donovan’s conjecture for {mathcal {O}}-blocks is a consequence of conjectures predicting bounds on the {mathcal {O}}-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field.

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.

Robinson’s conjecture on heights of characters

Geoffrey Robinson conjectured in 1996 that the $p$-part of character degrees in a $p$-block of a finite group can be bounded in terms of the center of a defect group of the block. We prove this conjecture for all primes $p\neq 2$ for all finite groups. Our argument relies on a reduction by Murai to the case of quasi-simple groups which are then studied using deep results on blocks of finite reductive groups.

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.

Robinson’s conjecture for classical groups

Abstract Robinson’s conjecture states that the height of any irreducible ordinary character in a block of a finite group is bounded by the size of the central quotient of a defect group. This conjecture had been reduced to quasi-simple groups by Murai. The case of odd primes was settled completely in our predecessor paper. Here we investigate the 2-blocks of finite quasi-simple classical groups.

The Navarro–Tiep Galois conjecture for $$p=2$$ p = 2

We prove a recent conjecture of Navarro and Tiep on the 2-rationality of characters of finite groups in relation with the structure of the commutator factor group of a Sylow 2-subgroup. On the way we complete the characterisation of characters of odd degree in quasi-simple groups of Lie type.

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.

Donovan’s conjecture and blocks with abelian defect groups

We give a reduction of Donovan's conjecture for abelian groups to a similar statement for quasisimple groups. Consequently we show that Donovan's conjecture holds for abelian $2$-groups.

Bases for quasisimple linear groups

Let $V$ be a vector space of dimension $d$ over $F_q$, a finite field of $q$ elements, and let $G \le GL(V) \cong GL_d(q)$ be a linear group. A base of $G$ is a set of vectors whose pointwise stabiliser in $G$ is trivial. We prove that if $G$ is a quasisimple group (i.e. $G$ is perfect and $G/Z(G)$ is simple) acting irreducibly on $V$, then excluding two natural families, $G$ has a base of size at most 6. The two families consist of alternating groups ${\rm Alt}_m$ acting on the natural module of dimension $d = m-1$ or $m-2$, and classical groups with natural module of dimension $d$ over subfields of $F_q$.