Recognizing direct products from their conjugate type vectors
type: Item
instance of: scholarly article
title: Recognizing direct products from their conjugate type vectors
zbMATH Open document ID: 0968.20017
published in: Journal of Algebra
publication date: 2001-08-27T00:00:00Z
review text: Let \(x\) be an element of a finite group \(G\); then \(|G:C_G(x)|\) is called the index of \(x\) in \(G\). Let \(n_r>\cdots>n_2>n_1=1\) denote all the distinct indices of elements of \(G\). Then \((n_r,\dots,n_2,n_1)\) is called the conjugate type vector of \(G\). If, in addition, \((m_s,\dots,m_2,m_1)\) is the conjugate type vector of \(H\), then \((n_r,\dots,n_2,n_1)\times(m_s,\dots,m_2,m_1)=(n_im_j\mid 1\leq i\leq r,\;1\leq j\leq s)\) is, after reordering of members, the conjugate type vector of \(G\times H\). Main result of this note is the following Theorem 1. Suppose \(G\) has conjugate type vector \((p^{a_s}_s,\dots,p^{a_1}_1,1)\times(q^{b_r}_r,\dots,q^{b_1}_1,1)\), where \(p_s,\dots,p_1\), \(q_r,\dots,q_1\) are distinct primes. Then \(r,s\leq 2\) and \(G=A\times B\), where \(A\) has conjugate type vector \((p^{a_s}_s,\dots,p^{a_1}_1,1)\) and \(B\) has conjugate type vector \((q^{b_r}_r,\dots,q^{b_1}_1,1)\).
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 1565875
zbMATH Keywords: finite groups
zbMATH Keywords: conjugate type vectors
author: A. R. Camina
author: Rachel D. Camina
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
OpenAlex ID: W2032010322
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DOI: 10.1006/JABR.2000.8535
description: scientific article; zbMATH DE number 1565875