Finite \(p\)-groups with a minimal non-Abelian subgroup of index \(p\). I.
Let \(G\) be a nonabelian \(p\)-group with \(p\) odd. \(G\) is called an \(A_k\)-group if each of its subgroups of index \(p^k\) is Abelian but this does not hold for subgroups of index \(p^{k-1}\). The problem of classifying \(A_2\)-groups has been solved recently by \textit{Q. Zhang, X. Sun, L. An} and \textit{M. Xu} [in Algebra Colloq. 15, No. 1, 167-180 (2008; Zbl 1153.20018)]. This is the first part of a series of papers solving the following problem due to Berkovich: Classify groups \(G\) containing an \(A_1\)-subgroup of index \(p\). The question is split into two parts, in the present paper the simpler one is solved as follows: the group \(G\) contains a unique \(A_1\)-subgroup of index \(p\) if and only if either it is such a group of order \(3^5\); or it is of order greater than \(p^5\), it has no Abelian maximal subgroup and has a maximal section which is a 3-group of maximal class and has an Abelian maximal subgroup. Moreover, isomorphism classes of these 3-groups \(G\) are completely described by generators and relations.
scientific article; zbMATH DE number 6104621
scientific article