Subgroups of powerful groups.
A finite \(p\)-group \(G\) is said to be powerful if \(p>2\) and \(G'\leq G^p\) or \(p=2\) and \(G'\leq G^4\), where \(G^n\) denotes the subgroup generated by all \(n\)-th powers. In this paper the authors give a new proof of Lubotzky's theorem [\textit{J. D. Dixon, M. P. F. du Sautoy, A. Mann} and \textit{D. Segal}, Analytic pro-\(p\)-groups (1999; Zbl 0934.20001)]. However, the question of which \(p\)-groups appear as subgroups of powerful groups is still open. The authors prove some results about the structure of 2-generator groups which occur as normal subgroups of powerful groups. Also they prove the following results: A finite \(p\)-group \(H\) can be embedded in a powerful group \(G\) such that \(G=H\cdot Z(G)\) if and only if \(H/Z(H)\) is powerful. If \(H\) is a \(p\)-group of class two then \(H\) can be embedded in a powerful group. If \(G\) is a \(p\)-group, \(p>2\), with a cyclic commutator subgroup then \(G\) can be embedded as a maximal subgroup of a powerful \(p\)-group.
scientific article; zbMATH DE number 2060050