Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree
Mathematics and Computer Science
Volume 1, Issue 1, May 2016, Pages: 17-20
Received: Apr. 11, 2016;
Accepted: May 3, 2016;
Published: May 28, 2016
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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
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For a finite group G
, we write
to denote the prime divisor set of the various conjugacy class lengths of G
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
a forest, the inequality
Prime Graph, Conjugacy Class Length, Almost Simple Group
To cite this article
Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree, Mathematics and Computer Science.
Vol. 1, No. 1,
2016, pp. 17-20.
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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