Control of transfer for odd primes

Abstract

Suppose p is a prime and T is a non-identity finite p-group. We show that if p is odd, then there exist two non-identity characteristic subgroups K1(T) and K2(T) of T such that K1 and K2 jointly control transfer in every finite group G containing T as a Sylow p-subgroup, in the sense that the normalizers of K1(T) and K2(T) in G determine the largest abelian factor group of G that is a p-group. We also give a new example of a characteristic subgroup K(T) of T such that K controls transfer in G by itself if p⩾5.