599 research outputs found
Infinite products of finite simple groups
We classify those sequences of
finite simple nonabelian groups such that the full product
has property (FA).Comment: AMS-LaTex file, 44 pages. To appear in Tran. 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
Biperfect Hopf Algebras
Recall that a finite group is called perfect if it does not have non-trivial
1-dimensional representations (over the field of complex numbers C). By
analogy, let us say that a finite dimensional Hopf algebra H over C is perfect
if any 1-dimensional H-module is trivial. Let us say that H is biperfect if
both H and H^* are perfect. Note that, H is biperfect if and only if its
quantum double D(H) is biperfect. It is not easy to construct a biperfect Hopf
algebra of dimension >1. The goal of this note is to describe the simplest such
example we know. The biperfect Hopf algebra H we construct is based on the
Mathiew group of degree 24, and it is semisimple. Therefore, it yields a
negative answer to Question 7.5 from a previous paper of the first two authors
(math.QA/9905168). Namely, it shows that Corollary 7.4 from this paper stating
that a triangular semisimple Hopf algebra over C has a non-trivial group-like
element, fails in the quasitriangular case. The counterexample is the quantum
double D(H).Comment: 5 pages, late
Embeddings of Sz(32) in E_8(5)
We show that the Suzuki group Sz(32) is a subgroup of E_8(5), and so is its automorphism group. Both are unique up to conjugacy in E_8(F) for any field F of characteristic 5, and the automorphism group Sz(32):5 is maximal in E_8(5)
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