Groups with few class sizes and the centraliser equality subgroup.
type: Item
instance of: scholarly article
title: Groups with few class sizes and the centraliser equality subgroup.
zbMATH Open document ID: 1058.20023
DOI: 10.1007/BF02771541
author: Avinoam Mann
published in: Israel Journal of Mathematics
publication date: 2004-10-15T00:00:00Z
review text: Let \(G\) be a finite \(p\)-group and let \(G\) have conjugacy class sizes \(1=n_1<n_2<\cdots<n_k\). We refer to classes of size \(n_2\) as `minimal classes', and to their elements as `minimal elements'. If \(n_s=p^{b_s}\) and \(x\in G\) has \(n_s\) conjugates, we say that \(x\) has `breadth' \(b(x)=b_s\). The `breadth' of \(G\) is \(b(G)=b_k\). Let \(D(G)=\langle x\in G\mid C_G(x)=C_G(x^p)\rangle\). The subgroup \(D(G)\) is a characteristic Abelian subgroup of \(G\). If \(G\) has \(k\) conjugacy class sizes, then \(\exp(G/D(G))\leq p^{k-1}\). (N. Ito proved this for \(k=2\).) If \(p=2\) and \(k>2\), then \(\exp(G/D(G))\leq 2^{k-2}\). We have \(D(G)\geq Z(G)\). If \(\exp(G)=p\), then \(D(G)=Z(G)\). If \(G\) is a \(2\)-group of maximal class and order \(>2^3\), then \(D(G)\) is a cyclic subgroup of index \(2\) in \(G\). If \(D(G)\leq H\leq G\) and \(\text{cl}(H)\leq p\), then \(D(G)\leq Z(H)\). Next, \(C_G(D(G))\geq Z_p(G)\), the \(p\)-th member of the upper central series of \(G\). The subgroup \(C_G(D)\) contains all normal subgroups of \(G\) of class \(<p\) and all elements of breadth \(<p\). In particular, \(C_G(D(G))\) contains all maximal Abelian normal subgroups of \(G\). The above results show the important role of the new characteristic subgroup \(D(G)\). A \(p\)-group \(G\) that is generated by its minimal classes, has class \(\leq 3\). In particular, if \(k=2\), then \(\text{cl}(G)\leq 3\) (K. Ishikawa). The subgroup, generated by the minimal classes of a \(2\)-group \(G\), is of class at most \(2\) and centralizes \(\mho_1(G)\). The subgroup, generated by the minimal classes of a metabelian \(p\)-group \(G\), is of class \(\leq 3\) and centralizes \([G,G,G]\). It is worthwhile to mention some further results presenting independent interest. Proposition 5. If \(G\) is a \(2\)-group, the minimal elements \(x\in G\) satisfying \(x^2\in Z(G)\), lie in \(Z_2(G)\). Proposition 8. Let \(B\) be an Abelian subgroup of maximal order in a metabelian \(p\)-group \(G\…
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 2108127
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: conjugacy class sizes
zbMATH Keywords: breadths
zbMATH Keywords: characteristic subgroups
zbMATH Keywords: upper central series
zbMATH Keywords: normal subgroups
zbMATH Keywords: centralizers
zbMATH Keywords: metabelian \(p\)-groups
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2108127