Finite groups whose Abelian subgroups have consecutive orders
type: Item
instance of: scholarly article
title: Finite groups whose Abelian subgroups have consecutive orders
zbMATH Open document ID: 0913.20015
author: Naoki Chigira
published in: Osaka Journal of Mathematics
publication date: 1999-03-01T00:00:00Z
review text: Let \(n\) be a positive integer. The finite group is called an \(OA_n\)-group (resp. an \(OC_n\)-group) if the possible orders of its Abelian (resp. cyclic) subgroups are \(\{1,2,\dots,n\}\), where it is assumed that all orders actually occur. \textit{R. Brandl} and \textit{W. Shi} [J. Algebra 143, No. 2, 388-400 (1991; Zbl 0745.20022)] have shown that \(OC_n\)-groups exist if and only if \(n\leq 8\). Using similar methods, the following is shown: Theorem. Let \(G\) be an \(OA_n\)-group. Then \(n\leq 6\) and \(G\) is isomorphic to \(S_k\) (\(k\leq 5\)) or the alternating groups \(A_4\), \(A_5\). In particular, there exist only finitely many \(OA_n\)-groups. The proof uses properties of the prime graph introduced by \textit{J. S. Williams} [J. Algebra 69, 487-513 (1981; Zbl 0471.20013)] and depends on the classification of all finite simple groups.
reviewed by: Rolf Brandl
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20E34
zbMATH DE Number: 1217813
zbMATH Keywords: element orders
zbMATH Keywords: Abelian subgroups
zbMATH Keywords: cyclic subgroups
zbMATH Keywords: finite groups
zbMATH Keywords: \(OC_n\)-groups
zbMATH Keywords: \(OA_n\)-groups
zbMATH Keywords: prime graphs
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1217813