Finite \(p\)-groups with small subgroups generated by two conjugate elements.
type: Item
instance of: scholarly article
title: Finite \(p\)-groups with small subgroups generated by two conjugate elements.
zbMATH Open document ID: 1270.20017
author: Heng Lü
author: Wei Zhou
author: Hai Jing Xu
published in: Archiv der Mathematik
publication date: 2013-05-07T00:00:00Z
review text: A \(p\)-group \(G\) is said to be a \(K_1\)-group if \(|\langle x,x^y\rangle:\langle x\rangle|\leq p\) for all \(x,y\in G\). The main result is the following: Theorem. Let a \(p\)-group \(G\) be a \(K_1\)-group. (a) If \(p\geq 3\), then \(\mho_1(G)\leq Z(G)\) and \(\exp(G')=p\). If, in addition, \(p=3\), then \(\mathrm{cl}(G)\leq 4\). If \(p\geq 5\), then \(\mathrm{cl}(G)\leq 3\). (b) If \(p=2\), then \(\mho_2(G)\leq\text Z(G)\), \(\exp(G')\leq4\) and \(\mathrm{cl}(G)\leq 3\). Also the following proper subclass \(K_2\) of the class \(K_1\) is considered. A \(p\)-group \(G\) is said to be a \(K_2\)-group if \(|\langle H,H^x \rangle:\langle H\rangle|\leq p\) for all \(x\in G\) and \(H\leq G\). If \(G\) is a \(K_2\)-group of exponent \(p\), then \(|G'|\leq p\). In general case, \(\mathrm{cl}(G)\leq 3\), and if \(p\geq 3\), then \(G'\) is elementary Abelian.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 6161243
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: conjugate elements
zbMATH Keywords: nilpotency class
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: s00013-012-0478-8
OpenAlex ID: W2090244987
cites work: Groups of prime power order. Vol. 1.
cites work: Groups of prime power order. Vol. 3.
cites work: Two classes of metabelian p-groups
cites work: On generalized Dedekind groups and Tarski super monsters
cites work: Q5840069
cites work: Some finite \(p\)-groups with bounded index of every cyclic subgroup in its normal closure.
DOI: 10.1007/S00013-012-0478-8
description: scientific article; zbMATH DE number 6161243