Bounding the index of the agemo in finite \(p\)-groups.
Let \(G\) be a finite \(p\)-group, \(p\) an odd prime number, \(p^k\) the maximal order of an exponent \(p\) subgroup. \textit{A. Mann} [J. Algebra 318, No. 2, 953-956 (2007; Zbl 1134.20017)] has shown that the index of the subgroup \(G^p\) is bounded by \(p^{k(\log_{2}k+2)}\). The author, in the main result of the paper, improves this bound to \(p^{k\log_{p-1}(k+1)}\) provided \(p^k\) is the maximal order of an exponent \(p\) subgroup of class smaller than \(p\). In the course of the proof the following notions are defined: given the set of words \(\mathcal V=\{w_i(x_1,\dots,x_{k_i})\mid i=1,\dots,s\}\) a verbal subgroup is \[ \mathcal V(G)=\langle w_1(x_1,\dots,x_{k_1}),\dots,w_s(x_1,\dots,x_{k_s})\mid x_i\in G\rangle. \] The group \(G\) is said to be \(\mathcal V\)-maximal if for all proper subgroups \(H\) of \(G\), \(|H:\mathcal V(H)|<|G:\mathcal V(G)|\). Let the \(D_l\) be dimension subgroups of \(G\) (\(2\leq l\leq p-1\)), these are verbal subgroups. It is proved that if the group \(G\) is \(D_l\)-maximal then it of class at most \(l\). In the proof of the main result the notion of a PF-group is also used [\textit{G. A. Fernández-Alcober, J. González-Sánchez, A. Jaikin-Zapirain}, Isr. J. Math. 166, 393-412 (2008; Zbl 1153.20028)], and it is shown that if \(G\) is a PF-group in which elements of order \(p\) form a subgroup then the index of \(G^p\) and the order of the subgroup \(\Omega_1(G)\) coincide. Some further results concerning the power structure are given, such as: if \(s\) is the upper integral part of \(k+1\over p-1\) and \(N\) is a normal subgroup contained in \(D_s\) then \(\Omega_i(N)^{p^i}=1\).
scientific article; zbMATH DE number 5655728
scientific article