Generation of finite almost simple groups by conjugates

By Robert M. Guralnick, Jan Saxl. 2003

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Let G be a finite almost simple group, with L = F ∗(G) the unique minimal norma subgroup ofG—soL is a finite simple group. For x ∈G, letα(x) be the minimal numbe of L-conjugates of x which generate the group 〈L,x〉. In this paper we obtain some upp bounds onα(x). Of course, this number depends on the type of L and onx. If G = Sn, then it is easy to see that α(x)= n− 1 for x a transposition and α(x) n− 1 for all nontrivial x. (See Lemma 6.1 for an improved easy estimate and [30, Theorem 4] for fu results.) Similarly, ifG is a classical group of dimension n, then one needs at least n conjugates of a pseudoreflection (here a pseudoreflection is an element whose fixed space i hyperplane). One result we shall prove is that if n 5 thenα(x) n, except for symplectic groups in even characteristic, where α(x) n+ 1. More generally, ifL is a finite simple group of Lie type of untwisted rank thanα(x) + 3, with a few explicitly determined exceptions. (Recall that the untwisted rank is the rank of the ambient algebraic g Our results are improvements on the results obtained by Hall et al. in [30]: they prov boundα(x) 8(2 + 1) in general, andα(x) 2 + 1 if x is a long root element. One motivation for studying this problem is to classify groups acting irreducibly primitively on a vector spaceV containing an element x such that dim[x,V ] is small compared to the dimension of V . This problem was considered by Gordeev [24] characteristic zero, and by Hall et al. [30] in general. One can show that a conseq of the bounds in [30] is the result, for both algebraic and finite groups, that any prim

Subfields: Artificial Intelligence, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

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Citations in the corpus

Rational irreducible characters and rational conjugacy classes in finite groups (2007, background): “By [9], we have that three conjugates of g generate S.”
Local-global principles for representations of quadratic forms (2006, methodology): “From [ 22, Theorem 8.4 ], we know that one of the following holds:”
Symplectic reflection algebras (2007, background): “The good news is that indecomposable symplectic reflection groups were classified by Huffmann– Wales, [95], see also Cohen, [37], and Guralnick–Saxl, [87].”