79 research outputs found
On base sizes for actions of finite classical groups
Let G be a finite almost simple classical group and let
? be a faithful primitive non-standard G-set. A base for G is a subset B C_ ? whose pointwise stabilizer is trivial; we write b(G) for the minimal size of a base for G. A well-known conjecture of Cameron and Kantor asserts that there exists an absolute constant c such that b(G) ? c for all such groups G, and the existence of such an undetermined constant has been established by Liebeck and Shalev. In this paper we prove that either b(G) ? 4, or G = U6(2).2, G? = U4(3).22 and b(G) = 5.
The proof is probabilistic, using bounds on fixed point ratios
On the prime graph of simple groups
Let be a finite group, let be the set of prime divisors of
and let be the prime graph of . This graph has vertex set
, and two vertices and are adjacent if and only if contains
an element of order . Many properties of these graphs have been studied in
recent years, with a particular focus on the prime graphs of finite simple
groups. In this note, we determine the pairs , where is simple and
is a proper subgroup of such that .Comment: 11 pages; to appear in Bull. Aust. Math. So
On the uniform domination number of a finite simple group
Let be a finite simple group. By a theorem of Guralnick and Kantor,
contains a conjugacy class such that for each non-identity element , there exists with . Building on this deep
result, we introduce a new invariant , which we call the uniform
domination number of . This is the minimal size of a subset of conjugate
elements such that for each , there exists with . (This invariant is closely related to the total
domination number of the generating graph of , which explains our choice of
terminology.) By the result of Guralnick and Kantor, we have for some conjugacy class of , and the aim of this paper
is to determine close to best possible bounds on for each family
of simple groups. For example, we will prove that there are infinitely many
non-abelian simple groups with . To do this, we develop a
probabilistic approach, based on fixed point ratio estimates. We also establish
a connection to the theory of bases for permutation groups, which allows us to
apply recent results on base sizes for primitive actions of simple groups.Comment: 35 pages; to appear in Trans. Amer. Math. So
On the involution fixity of exceptional groups of Lie type
The involution fixity of a permutation group of degree
is the maximum number of fixed points of an involution. In this paper we
study the involution fixity of primitive almost simple exceptional groups of
Lie type. We show that if is the socle of such a group, then either , or and is a Suzuki
group in its natural -transitive action of degree . This bound is
best possible and we present more detailed results for each family of
exceptional groups, which allows us to determine the groups with . This extends recent work of Liebeck and Shalev, who
established the bound for every almost simple
primitive group of degree with socle (with a prescribed list of
exceptions). Finally, by combining our results with the Lang-Weil estimates
from algebraic geometry, we determine bounds on a natural analogue of
involution fixity for primitive actions of exceptional algebraic groups over
algebraically closed fields.Comment: 45 pages; to appear in Int. J. Algebra Compu
Large subgroups of simple groups
Let be a finite group. A proper subgroup of is said to be large
if the order of satisfies the bound . In this note we
determine all the large maximal subgroups of finite simple groups, and we
establish an analogous result for simple algebraic groups (in this context,
largeness is defined in terms of dimension). An application to triple
factorisations of simple groups (both finite and algebraic) is discussed.Comment: 37 page
Fixed point spaces in primitive actions of simple algebraic groups
AbstractLet G be a simple algebraic group of adjoint type acting primitively on an algebraic variety Ī©. We study the dimensions of the subvarieties of fixed points of involutions inĀ G. In particular, we obtain a close to best possible function f(h), where h is the Coxeter number of G, with the property that with the exception of a small finite number of cases, there exists an involution t in G such that the dimension of the fixed point space of t is at least f(h)dimĪ©
Base sizes for primitive groups with soluble stabilisers
Let be a finite primitive permutation group on a set with point
stabiliser . Recall that a subset of is a base for if its
pointwise stabiliser is trivial. We define the base size of , denoted
, to be the minimal size of a base for . Determining the base size
of a group is a fundamental problem in permutation group theory, with a long
history stretching back to the 19th century. Here one of our main motivations
is a theorem of Seress from 1996, which states that if
is soluble. In this paper we extend Seress' result by proving that for all finite primitive groups with a soluble point
stabiliser . This bound is best possible. We also determine the exact base
size for all almost simple groups and we study random bases in this setting.
For example, we prove that the probability that random elements in
form a base tends to as tends to infinity.Comment: 43 pages; to appear in Algebra and Number Theor
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