Critical maximal subgroups and conjugacy of supplements in finite soluble groups
نویسنده
چکیده
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 subgroups and 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 (Theorem 1 of [3]). Furthermore, we consider families of maximal subgroups of G closed under conjugation whose intersection equals Φ(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, these are exactly
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