1,067 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
Irreducible subgroups of simple algebraic groups - a survey
Let be a simple linear algebraic group over an algebraically closed field
of characteristic , let be a proper closed subgroup of
and let be a nontrivial finite dimensional irreducible rational
-module. We say that is an irreducible triple if is
irreducible as a -module. Determining these triples is a fundamental
problem in the representation theory of algebraic groups, which arises
naturally in the study of the subgroup structure of classical groups. In the
1980s, Seitz and Testerman extended earlier work of Dynkin on connected
subgroups in characteristic zero to all algebraically closed fields. In this
article we will survey recent advances towards a classification of irreducible
triples for all positive dimensional subgroups of simple algebraic groups.Comment: 31 pages; to appear in the Proceedings of Groups St Andrews 201
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 base sizes for algebraic groups
Let be a permutation group on a set . A subset of is a
base for if its pointwise stabilizer is trivial; the base size of is
the minimal cardinality of a base. In this paper we initiate the study of bases
for algebraic groups defined over an algebraically closed field. In particular,
we calculate the base size for all primitive actions of simple algebraic
groups, obtaining the precise value in almost all cases. We also introduce and
study two new base measures, which arise naturally in this setting. We give an
application concerning the essential dimension of simple algebraic groups, and
we establish several new results on base sizes for the corresponding finite
groups of Lie type. The latter results are an important contribution to the
classical study of bases for finite primitive permutation groups. We also
indicate some connections with generic stabilizers for representations of
simple algebraic groups.Comment: 62 pages; to appear in J. Eur. Math. Soc. (JEMS
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
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