On the structure of normal subgroups of potent \(p\)-groups.
Let \(G\) be a finite \(p\)-group. We say that \(G\) is `potent' if \(G'\leq\mho^2(G)\) for \(p=2\) or \(K_{p-1}(G)\leq\mho_1(G)\) for \(p>2\) (here \(K_n(G)\) is the \(n\)-th member of the lower central series of \(G\)). If \(p<5\), then potent \(p\)-groups coincide with powerful \(p\)-groups. Clearly, powerful \(p\)-groups are potent. A \(p\)-group \(G\) is said to be `power Abelian', if, for all \(i\in\mathbb{N}\), we have \(\mho_i(G)=\{g^{p^i}\mid g\in G\}\). \(\Omega_i(G)=\{x\in G\mid o(g)\leq p^i\}\) and \(|\mho_i(G)|=|G:\Omega_i(G)|\). Main Theorem. Let \(G\) be a potent group. (1) If \(p=2\), then (a) \(\exp(\Omega_i(G))\leq 2^{i+1}\) and, even more, \(\mho_i([\Omega_i(G),G])=\mho_i(\Omega_i(\mho_1(G)))=\{1\}\); (b) the class of \(\Omega_i(G)\) is at most \(\tfrac12(i+3)\); (c) if \(N\) is normal in \(G\) and \(N\leq\mho_1(G)\), then \(N\) is power Abelian; (d) if \(N\) is normal in \(G\) and \(N\leq\mho_2(G)\), then \(N\) is powerful. (2) If \(p>2\), then (a) \(\exp(\Omega_i(G))\leq p^i\); (b) the class of \(\Omega_i(G)\) is at most \((p-2)i+1\); (c) if \(N\) is normal in \(G\), then \(N\) is power Abelian; (d) if \(N\) is normal in \(G\) and \(N\leq\mho_1(G)\), then \(N\) is powerful. As we see, the theorem yields additional useful information on the normal structure of powerful \(p\)-groups. In conclusion, two examples are presented. In the first example, a powerful \(2\)-group \(G\) is given such that \(\exp(\Omega_i(G))=2^{i+1}\) and \(G\) contains a normal subgroup \(N\) such that \(\mho_1(N)\neq\{x^2\mid x\in N\}\). In the second example, a powerful \(2\)-group \(G\) has a normal subgroup \(N\) such that \(\mho_1(N)'\nleq\mho_2(N')\).
scientific article; zbMATH DE number 2105412