2,723 research outputs found
Adequate Subgroups II
The notion of adequate subgroups was introduced by Jack Thorne. It is a
weakening of the notion of big subgroup used by Wiles and Taylor in proving
automorphy lifting theorems for certain Galois representations. Using this
idea, Thorne was able to prove some new lifting theorems. In an appendix to
Thorne's article, it was shown in that certain groups were adequate. One of the
key aspects was the question of whether the span of the semsimple elements in
the group is the full endomorphism ring of an absolutely irreducible module. We
show that this is the case in prime characteristic p for p-solvable groups as
long the dimension is not divisible by p. We also observe that the condition
holds for certain infinite groups. Finally, we present the first examples
showing that this condition need not hold and give a negative answer to a
question of Richard Taylor.Comment: to appear in Bulletin of Mathematical Science
On a subfactor generalization of Wall's conjecture
In this paper we discuss a conjecture on intermediate subfactors which is a
generalization of Wall's conjecture from the theory of finite groups. We
explore special cases of this conjecture and present supporting evidence. In
particular we prove special cases of this conjecture related to some finite
dimensional Kac Algebras of Izumi-Kosaki type which include relative version of
Wall's conjecture for solvable groups.Comment: 16 page
Products of conjugacy classes and fixed point spaces
We prove several results on products of conjugacy classes in finite simple
groups. The first result is that there always exists a uniform generating
triple. This result and other ideas are used to solve a 1966 conjecture of
Peter Neumann about the existence of elements in an irreducible linear group
with small fixed space. We also show that there always exist two conjugacy
classes in a finite non-abelian simple group whose product contains every
nontrivial element of the group. We use this to show that every element in a
non-abelian finite simple group can be written as a product of two rth powers
for any prime power r (in particular, a product of two squares).Comment: 44 page
Simple groups admit Beauville structures
We answer a conjecture of Bauer, Catanese and Grunewald showing that all
finite simple groups other than the alternating group of degree 5 admit unmixed
Beauville structures. We also consider an analog of the result for simple
algebraic groups which depends on some upper bounds for character values of
regular semisimple elements in finite groups of Lie type and obtain definitive
results about the variety of triples in semisimple regular classes with product
1. Finally, we prove that any finite simple group contains two conjugacy
classes C,D such that any pair of elements in C x D generates the group.Comment: 30 pages, in the second version, some results are improved and in
particular we prove an irreducibility for a certain variet
- …