Critical maximal subgroups and conjugacy of supplements in finite soluble groups

Abstract

Abstract Let G be a finite group with an abelian normal subgroup N. When does N have a unique conjugacy class of complements in G? We consider this question with a focus on properties of maximal subgroups. As corollaries we obtain Theorems 1.6 and 1.7 which are closely related to a result by Parker and Rowley on supplements of a nilpotent normal subgroup [3, Theorem 1]. Furthermore, we consider families of maximal subgroups of G closed under conjugation whose intersection equals Φ ⁢ ( G ) {\Phi(G)} . In particular, we characterize the soluble groups having a unique minimal family with this property (Theorem 2.3, Remark 2.4). In the case when Φ ⁢ ( G ) = 1 {\Phi(G)=1} , these are exactly the soluble groups in which each abelian normal subgroup has a unique conjugacy class of complements.