Conjugacy conditions for supersoluble complements of an abelian base and a fixed point result for non-coprime actions

We demonstrate that two supersoluble complements of an abelian base in a finite split extension are conjugate if and only if, for each prime p, a Sylow p -subgroup of one complement is conjugate to a Sylow p -subgroup of the other. As a corollary, we find that any two supersoluble complements of an abelian subgroup N in a finite split extension G are conjugate if and only if, for each prime p, there exists a Sylow p -subgroup S of G such that any two complements of S∩N in S are conjugate in G . In particular, restricting to supersoluble groups allows us to ease D. G. Higman's stipulation that the complements of S∩N in S be conjugate within S. We then consider group actions and obtain a fixed point result for non-coprime actions analogous to Glauberman's lemma.