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

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