Adequate Subgroups II

TL;DR

This paper explores the properties of adequate subgroups, extending Thorne's work, and provides new results on when certain groups satisfy adequacy conditions, including counterexamples to previous assumptions.

Contribution

It proves that prime characteristic p groups are adequate under specific conditions and presents the first examples where adequacy fails, answering a question by Richard Taylor.

Findings

Prime characteristic p groups are adequate if the dimension is not divisible by p.

Certain infinite groups satisfy the adequacy condition.

Counterexamples show adequacy condition does not always hold.

Abstract

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.