Subgroups of Maximal Rank in Finite Exceptional Groups of Lie Type
Proceedings of the London Mathematical Society, 1992
The purpose of this paper is to investigate a large and natural class of maximal subgroups of the... more The purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of Lie type, which we call subgroups of maximal rank. These subgroups play a prominent role both in the classification of local maximal subgroups in [9, Theorem ...
Linear Groups with Orders Having Certain Large Prime Divisors
Proceedings of the London Mathematical Society, 1999
Page 1. LINEAR GROUPS WITH ORDERS HAVING CERTAIN LARGE PRIME DIVISORS ROBERT GURALNICK, TIM PENTT... more Page 1. LINEAR GROUPS WITH ORDERS HAVING CERTAIN LARGE PRIME DIVISORS ROBERT GURALNICK, TIM PENTTILA, CHERYL E. PRAEGER and JAN SAXL [Received 14 February 1997ÐRevised 20 October 1997] Abstract ...
This paper represents the first step in the classification of finite primitive distance transitiv... more This paper represents the first step in the classification of finite primitive distance transitive graphs. In it we reduce the problem to the case where the automorphism group is either almost simple or affine. Let^ be a simple, connected, undirected graph with vertex set Q. If oc,/? e Q, then d (a, j8) denotes the distance between a and/3 in (§ . Let G be some group of automorphisms of c § .
On the Overgroups of Irreducible Subgroups of the Finite Classical Groups
Proceedings of the London Mathematical Society, 1987
Let Xo = Xo{V) be a finite simple classical group with natural (projective) module V over GF(pa),... more Let Xo = Xo{V) be a finite simple classical group with natural (projective) module V over GF(pa), a finite field of prime characteristic p. Denote by Xx the full classical group on V corresponding to Xo (so, for example, if Xo = PSLn(pa) then Xx = ?rLn(pa)), and let X be a group such ...
Subgroups of Algebraic Groups Containing Regular Unipotent Elements
Journal of the London Mathematical Society, 1997
ABSTRACT Let G be a simple algebraic group over an algebraically closed field K of characteristic... more ABSTRACT Let G be a simple algebraic group over an algebraically closed field K of characteristic p with p≥0. Then G contains finitely many conjugacy classes of unipotent elements. There is a unique class of unipotent elements of largest dimension – the class of regular unipotent elements; the centralizer of any such element is a unipotent group of dimension equal to the rank of G. For example, if G is the linear group SL(V) on the finite-dimensional vector space V over K, then the regular unipotent elements are precisely the unipotent matrices consisting of a single Jordan block.
In (LS), Landazuri and Seitz gave lower bounds for irreducible representations of Chevalley group... more In (LS), Landazuri and Seitz gave lower bounds for irreducible representations of Chevalley groups in non-defining characteristic (when referring to irreducible repre- sentations for quasisimple groups G, we will assume that the modules are nontrivial on F (G)). See also (SZ), (GPPS), (Ho) for some improvements on these bounds. These results have proved to be useful in many applications. In
Each finite simple group other than U3(3 ) can be generated by three of its involutions. In fact,... more Each finite simple group other than U3(3 ) can be generated by three of its involutions. In fact, each such group is generated by two elements, of which one is strongly real and the other is an involution.
This paper is a contribution to the programme to classify finite distance-transitive graphs and t... more This paper is a contribution to the programme to classify finite distance-transitive graphs and their automorphism groups. We classify pairs ( , G) where is a graph and G is an automorphism group of acting distance-transitively and primitively on the vertex set of , subject to the condition that there is a normal elementary abelian subgroup V in G which acts regularly on the vertex set of and the stabilizer G 0 of a vertex (which is a complement to V in G) has a unique non-abelian composition factor isomorphic to one of the 26 sporadic simple groups. There are exactly 10 examples of , all known for a long time.
Distance-transitive representations of the sporadic groups
Communications in Algebra, 1995
... a finite vector space, and the stabilizer of a point acts as an irreducible linear group on V... more ... a finite vector space, and the stabilizer of a point acts as an irreducible linear group on V. The second class are almost simple permutation groups. In this case G is a nonabelian simple group, possibly extended by outer automorphisms. So the classification of primitive distance ...
A base of a permutation group G on a set Ω is a subset B of Ω such that the pointwise stabilizer ... more A base of a permutation group G on a set Ω is a subset B of Ω such that the pointwise stabilizer of B in G is trivial. The base size of G, denoted by b(G), is the minimal cardinality of a base. Let G = Sn or An acting primitively on a set with point stabilizer H. In this note we prove that if H acts primitively on {1, . . . , n}, and does not contain An, then b(G) = 2 for all n ≥ 13. Combined with a theorem of James, this completes the classification of primitive actions of alternating and symmetric groups which admit a base of size two.
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