The notion of adequate subgroups was introduced by Jack Thorne [42]. It is a
weakening of the notion of big subgroups used in generalizations of the
Taylor-Wiles method for proving the automorphy of certain Galois
representations. Using this idea, Thorne was able to strengthen many automorphy
lifting theorems. It was shown in [22] that if the dimension is small compared
to the characteristic then all absolutely irreducible representations are
adequate. Here we extend the result by showing that, in almost all cases,
absolutely irreducible kG-modules in characteristic p, whose irreducible
G+-summands have dimension less than p (where G+ denotes the subgroup of G
generated by all p-elements of G), are adequate.Comment: 60 page
