Classical Groups Of Lie Type Research Articles

Word images in symmetric and classical groups of Lie type are dense

Let $w\in\mathbf F_k$ be a non-trivial word and denote by $w(G)\subseteq G$ the image of the associated word map $w\colon G^k\to G$. Let $G$ be one of the finite groups ${\rm S}_n,{\rm GL}_n(q),{\rm Sp}_{2m}(q),{\rm GO}_{2m}^\pm(q),{\rm GO}_{2m+1}(q),{\rm GU}_n(q)$ ($q$ a prime power, $n\geq 2$, $m\geq 1$), or the unitary group ${\rm U}_n$ over $\mathbb C$. Let $d_G$ be the normalized Hamming distance resp. the normalized rank metric on $G$ when $G$ is a symmetric group resp. one of the other classical groups and write $n(G)$ for the permutation resp. Lie rank of $G$. For $\varepsilon>0$, we prove that there exists an integer $N(\varepsilon,w)$ such that $w(G)$ is $\varepsilon$-dense in $G$ with respect to the metric $d_G$ if $n(G)\geq N(\varepsilon,w)$. This confirms metric versions of a conjectures by Shalev and Larsen. Equivalently, we prove that any non-trivial word map is surjective on a metric ultraproduct of groups $G$ from above such that $n(G)\to\infty$ along the ultrafilter. As a consequence of our methods, we also obtain an alternative proof of the result of Hui-Larsen-Shalev that $w_1({\rm SU}_n)w_2({\rm SU}_n)={\rm SU}_n$ for non-trivial words $w_1,w_2\in\mathbf F_k$ and $n$ sufficiently large.

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.

Recognition of PSL(2,p) ×PSL(2,p) by its complex group algebra

In [H. P. Tong-Viet, Simple classical groups of Lie type are determined by their character degrees, J. Algebra 357 (2012) 61–68] the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras? It is proved that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of [Formula: see text], the complex group algebra of [Formula: see text]. One of the next natural groups to be considered are the characteristically simple groups. In this paper, as the first step in this investigation we prove that if [Formula: see text] is an odd prime number, then [Formula: see text] is uniquely determined by the structure of its complex group algebra.

A new characterization for some extensions of PSL(2, q) for some q by some character degrees

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2,p2) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2,p2) of order 2|PSL(2,p2)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let p be an odd prime number and q = p or q = p3. Let M be a finite group such that |M| = h|PSL(2,q), where h is a divisor of |Out(PSL(2,q))|. Also suppose that M has an irreducible character of degree q and 2p does not divide the degree of any irreducible character of M. As the main result of this paper we prove that M has a unique nonabelian composition factor which is isomorphic to PSL(2,q). As a consequence of our result we prove that M is uniquely determined by its order and some information on its character degrees which implies that M is uniquely determined by the structure of its complex group algebra.

A supercharacter theory for involutive algebra groups

If J is a finite-dimensional nilpotent algebra over a finite field k, the algebra group P=1+J admits a (standard) supercharacter theory as defined in [16]. If J is endowed with an involution σ, then σ naturally defines a group automorphism of P=1+J, and we may consider the fixed point subgroup CP(σ)={x∈P:σ(x)=x−1}. Assuming that k has odd characteristic p, we use the standard supercharacter theory for P to construct a supercharacter theory for CP(σ). In particular, we obtain a supercharacter theory for the Sylow p-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by André and Neto in [7,8] for the special case of the symplectic and orthogonal groups.

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

Let G be a finite group. Denote by Irr(G) the set of all irreducible complex characters of G. Let cd(G) be the set of all irreducible complex character degrees of G forgetting multiplicities, that is, cd(G)={χ(1):χ∈Irr(G)} and let X1(G) be the set of all irreducible complex character degrees of G counting multiplicities. Let H be a finite nonabelian simple classical group. In this paper, we will show that if G is a finite group and X1(G)=X1(H) then G is isomorphic to H. In particular, this implies that the nonabelian simple classical groups of Lie type are uniquely determined by the structure of their complex group algebras.

Irreducible characters of groups associated with finite nilpotent algebras with involution

An algebra group is a group of the form P = 1 + J where J is a finite-dimensional nilpotent associative algebra. A theorem of Z. Halasi asserts that, in the case where J is defined over a finite field F , every irreducible character of P is induced from a linear character of an algebra subgroup of P . If ( J , σ ) is a nilpotent algebra with involution, then σ naturally defines a group automorphism of P = 1 + J , and we may consider the fixed point subgroup C P ( σ ) . Assuming that F has odd characteristic p , we show that every irreducible character of C P ( σ ) is induced from a linear character of a subgroup of the form C Q ( σ ) where Q is a σ -invariant algebra subgroup of P . As a particular case, the result holds for the Sylow p -subgroups of the finite classical groups of Lie type.

Sylow d-tori of classical groups and the McKay conjecture, II

The aim of this paper is to describe the irreducible ordinary characters of the normalisers of Sylow tori of a finite reductive group G of classical type. We prove that every character χ ∈ Irr ( L ) extends as an irreducible character to its inertia group in N , where L is the centraliser and N the normaliser of a Sylow torus in G . Using this result Malle has established in Malle (2007) [8] a bijection between Irr ℓ ′ ( G ) and Irr ℓ ′ ( N ) , where N is the normaliser of a suitable Sylow torus in G . This is a first step in proving the inductive McKay condition from Isaacs et al. (2007) [5] for the classical groups of Lie type. Furthermore the bijection enables us to prove the McKay conjecture for primes ℓ > 3 , different from the defining characteristic of G .

Minimal logarithmic signatures for finite groups of Lie type

A logarithmic signature (LS) for a finite group G is an ordered tuple ? = [A 1, A 2, . . . , A n ] of subsets A i of G, such that every element $${g \in G}$$ can be expressed uniquely as a product g = a 1 a 2 . . . a n , where $${a_i \in A_i}$$ . The length of an LS ? is defined to be $${l(\alpha)= \sum^{n}_{i=1}|A_i|}$$ . It can be easily seen that for a group G of order $${\prod^k_{j=1}{p_j}^{m_j}}$$ , the length of any LS ? for G, satisfies, $${l(\alpha) \geq \sum^k_{j=1}m_jp_j}$$ . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS) (Gonzalez Vasco et al., Tatra Mt. Math. Publ. 25:2337, 2002). The MLS conjecture states that every finite simple group has an MLS. This paper addresses the MLS conjecture for classical groups of Lie type and is shown to be true for the families PSL n (q) and PSp 2n (q). Our methods use Singer subgroups and the Levi decomposition of parabolic subgroups for these groups.

Equivalences of classifying spaces completed at the prime two

Introduction Higher limits over orbit categories Reduction to simple groups A relative version of $\Lambda$-functors Subgroups which contribute to higher limits Alternating groups Groups of Lie type in characteristic two Classical groups of Lie type in odd characteristic Exceptional groups of Lie type in odd characteristic Sproadic groups Computations of $\textrm{lim}^1(\mathcal{Z}_G)$ Bibliography.

A Combinatorial Generalization of the Springer Correspondence for Type Bn

We present a combinatorial generalization of the Springer correspondence of the classical groups of Lie type B and C. It is known that the Springer modules yield the tempered irreducible modules with real central character of a graded Hecke algebra of the same Lie type and with equal labels. We conjecture that our generalized correspondence provides the description of these modules in the case of unequal labels.

Irreducibility of tensor squares, symmetric squares and alternating squares

We investigate the question when the tensor square, the alternating square, or the symmetric square of an absolutely irreducible projective representation V of an almost simple group G is again irreducible. The knowledge of such representations is of importance in the description of the maximal subgroups of simple classical groups of Lie type. We show that if G is of Lie type in odd characteristic, either V is a Weil representation of a symplectic or unitary group, or G is one of a finite number of exceptions. For G in even characteristic, we derive upper bounds for the dimension of V which are close to the minimal possible dimension of nontrivial irreducible representations. Our results are complete in the case of complex representations. We will also answer a question of B. H. Gross about finite subgroups of complex Lie groups & that act irreducibly on all fundamental representations of &.

Irreducibility of alternating and¶symmetric squares

We investigate the question when the alternating or symmetric square of an absolutely irreducible projective representation of a non-abelian simple group G is again irreducible. The knowledge of such representations is of importance in the description of the maximal subgroups of simple classical groups of Lie type. We obtain complete results for G an alternating group and for G a projective special linear group when the given representation is in non-defining characteristic. For the proof we exhibit a linear composition factor in the socle of the restriction to a large subgroup of the alternating or symmetric square of a given projective representation V. Assuming irreducibility this shows that the dimension of V has to be very small. A good knowledge of projective representations of small dimension allows to rule out these cases as well.