Volume 1, Issue 1, May 2016, Page: 17-20
Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree
Liguo He, Dept. of Math., Shenyang University of Technology, Shenyang, PR China
Yaping Liu, Dept. of Math., Shenyang University of Technology, Shenyang, PR China
Jianwei Lu, Dept. of Math., Shenyang University of Technology, Shenyang, PR China
Received: Apr. 11, 2016;       Accepted: May 3, 2016;       Published: May 28, 2016
DOI: 10.11648/j.mcs.20160101.14      View  3057      Downloads  92
Abstract
For a finite group G, we write to denote the prime divisor set of the various conjugacy class lengths of G and the maximum number of distinct prime divisors of a single conjugacy class length of G. It is a famous open problem that can be bounded by . Let G be an almost simple group G such that the graph built on element orders is a tree. By using Lucido’s classification theorem, we prove except possibly when G is isomorphic to , where p is an odd prime and α is a field automorphism of odd prime order f. In the exceptional case, . Combining with our known result, we also prove that for a finite group G with a forest, the inequality is true.
Keywords
Prime Graph, Conjugacy Class Length, Almost Simple Group
To cite this article
Liguo He, Yaping Liu, Jianwei Lu, Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree, Mathematics and Computer Science. Vol. 1, No. 1, 2016, pp. 17-20. doi: 10.11648/j.mcs.20160101.14
Copyright
Copyright © 2016 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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