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CONJUGACY CLASS SIZE CONDITIONS WHICH IMPLY SOLVABILITY

Part of: Structure and classification of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  15 January 2013

QINGJUN KONG  and
QINGFENG LIU
QINGJUN KONG*
Affiliation:
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
QINGFENG LIU
Affiliation:
Department of Basic Sciences, Shandong Water Polytechnic, Rizhao 276826, China
*
✉[email protected]
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Abstract

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Let $G$ be a finite $p$-solvable group and let ${G}^{\ast } $ be the set of elements of primary and biprimary orders of $G$. Suppose that the conjugacy class sizes of ${G}^{\ast } $ are $\{ 1, {p}^{a} , n, {p}^{a} n\} $, where the prime $p$ divides the positive integer $n$ and ${p}^{a} $ does not divide $n$. Then $G$ is, up to central factors, a $\{ p, q\} $-group with $p$ and $q$ two distinct primes. In particular, $G$ is solvable.

Keywords

conjugacy class sizessolvable groupsfinite groups

MSC classification

Secondary: 20E45: Conjugacy classes 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 2 , October 2013 , pp. 297 - 300
Copyright
Copyright ©2012 Australian Mathematical Publishing Association Inc. 

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