On the structure of normal subgroups of potent \(p\)-groups.
type: Item
instance of: scholarly article
title: On the structure of normal subgroups of potent \(p\)-groups.
zbMATH Open document ID: 1056.20014
author: Andrei Jaikin-Zapirain
author: Jon Gonzalez-Sanchez
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: 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 \(\mho_1(N)'\nleq\mho_2(N')\).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 2105412
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: lower central series
zbMATH Keywords: potent \(p\)-groups
zbMATH Keywords: powerful \(p\)-groups
zbMATH Keywords: power Abelian \(p\)-groups
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.12.006
OpenAlex ID: W2046131805
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cites work: Powerful \(p\)-groups. I: Finite groups.
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DOI: 10.1016/J.JALGEBRA.2003.12.006
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description: scientific article; zbMATH DE number 2105412