Adequate subgroups II

Abstract

The notion of adequate subgroups was introduced by Thorne (J Inst Math Jussieu. arXiv:1107.5989, to appear). 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. It was shown in Guralnick et al. (J Inst Math Jussieu. arXiv:1107.5993, to appear) that certain groups were adequate. One of the key aspects was the question of whether the span of the semisimple 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.