\(d\)-maximal \(p\)-groups and Stiefel-Whitney classes of a regular representation
Let \(G\) be a \(p\)-group, \(\Phi(G)\) the Frattini subgroup of \(G\) and \(d(G)\) the minimal number of generators of \(G\). The group \(G\) is called \(d\)-maximal if \(d(K)<d(G)\) for every proper subgroup \(K\) of \(G\). [In J. Algebra 144, No. 1, 214-247 (1991; Zbl 0777.20019)] \textit{B. Kahn} introduced a numerical invariant for a 2-group \(G\) as \(\nu(G):=\min\{n>0\mid w_{2^{n-1}}(r_G)\neq 0\}\) where \(w_i(r_G)\) is the \(i\)th Stiefel-Whitney class of the regular representation \(r_G\) of \(G\), and conjectured that \(\nu(G)\geq d(G)\) proving that this is so if \(d(G) \leq 3\). His proof involves the study of \(d\)-maximal groups. Here the author shows that B. Kahn's conjecture holds if \(d(G)=4\) and obtains some interesting results concerning the nilpotence class, \(\text{cl}(G)\), of a \(d\)-maximal group \(G\), namely: Theorem 1. If \(G\) is \(d\)-maximal then \(\text{cl}(G)\leq 2\) and \(\inf:H^2(G/\Phi(G),\mathbb{Z}/p\mathbb{Z})\to H^2(G,\mathbb{Z}/p\mathbb{Z})\) provided that one of the following conditions is satisfied (a) \(p>2\), (b) \(p=2\) and \(|\Phi(G)|=2^m\) with \(m\leq 2\), (c) \(p=2\) and \(d(G)\leq 3\). Theorem 2. For every \(n\geq 4\), there exists a \(d\)-maximal 2-group \(K_n\) of class \(\geq 3\) with \(d(K_n)=n\). Note that it was shown by \textit{T. Laffey} [in Bull. Lond. Math. Soc. 5, 288-290 (1973; Zbl 0273.20013)] that every \(d\)-maximal \(p\)-group \(G\) with \(p>2\) has \(\text{cl}(G)\leq 2\).
scientific article; zbMATH DE number 843876
scientific article