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 $p$, Sylow $p$-subgroup one complement is to the other. As corollary, we find any subgroup $N$ $G$ there exists $S$ such $S\cap N$ $G$. In particular, restricting groups allows us ease D. G. Higman's stipulation be within $S$. then consider group actions obtain fixed point result non-coprime analogous Glauberman's lemma.