Let $G$ be a finite group and suppose that $P$ is a soluble $\{2, 3\}^\prime$ -subgroup of $G$ . The reader will lose only a little by assuming that $P$ is a subgroup of prime order $p > 3$ . Define \[ \Sigma_G(P) = \{A \le G \mid A \hbox{ is soluble and } A = \langle P, P^a\rangle\hbox{ for some }a \in A\}. \] This set is partially ordered by inclusion and we let \[ \Sigma^*_G(P) \] denote the set of maximal members of $\Sigma_G(P)$ .

If p is a prime, then a finite group is p-soluble if each of its composition factors is either a p-group or has order coprime to p. For example, soluble groups are p-soluble. However, there are many insoluble groups that are p-soluble. We shall prove the following result.