On finite 2-equilibrated \(p\)-groups (Q2629546)

The authors call a finite group \(G\) \(n\)-equilibrated if for all subgroups \(H\), \(K\) of \(G\) with \(d(H), d(K)\geq n\) either \(H\leqslant N_G(K)\), or \(K\leqslant N_G(H)\). Here, \(d(G)\) denotes the minimum number of generators of the group \(G\). In this technical paper, the authors begin the study of \(2\)-equilibrated \(p\)-groups. They first focus their attention on the case \(d(G)=2\) and completely classify the \(2\)-equilibrated \(p\)-groups \(G\) with \(d(G)=2\): they fall into seven families, which are described via presentations. In Section 4, additional information is obtained on \(2\)-equilibrated \(p\)-groups \(G\) with \(d(G)>2\). For example, if \(p\) is odd and if \(d(G)\geq 4\), then \(G\) is modular (Theorem 4.4). In Section 5, they study \(2\)-equilibrated \(2\)-groups \(G\) with \(d(G)\geq 4\) and they show that if also \(|G|\geq 2^7\) then \(G'\) is cyclic (Theorem 5.5). The last main result is Theorem 5.6, where the classification is given for all groups \(G\) such that \(G\) is a non-Dedekind \(2\)-equilibrated \(2\)-group of order \(2^n\) with \(n\geq 7\) and \(d(G)\geq 4\).