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ON $p$-PARTS OF CONJUGACY CLASS SIZES OF FINITE GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  28 March 2018

YONG YANG Open the ORCID record for YONG YANG [Opens in a new window]  and
GUOHUA QIAN
YONG YANG*
Affiliation:
Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX 78666, USA Key Laboratory of Group and Graph Theories and Applications, Chongqing University of Arts and Sciences, Chongqing, PR China email yang@txstate.edu
GUOHUA QIAN
Affiliation:
Department of Mathematics, Changshu Institute of Technology, Changshu, JiangSu 215500, PR China email ghqian2000@163.com
*
✉yang@txstate.edu
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Abstract

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Let $G$ be a finite group. Let $\operatorname{cl}(G)$ be the set of conjugacy classes of $G$ and let $\operatorname{ecl}_{p}(G)$ be the largest integer such that $p^{\operatorname{ecl}_{p}(G)}$ divides $|C|$ for some $C\in \operatorname{cl}(G)$. We prove the following results. If $\operatorname{ecl}_{p}(G)=1$, then $|G:F(G)|_{p}\leq p^{4}$ if $p\geq 3$. Moreover, if $G$ is solvable, then $|G:F(G)|_{p}\leq p^{2}$.

Keywords

conjugacy class sizes

MSC classification

Primary: 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure
Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 97 , Issue 3 , June 2018 , pp. 406 - 411
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The project was supported by NSFC (Nos. 11671063 and 11471054), the Natural Science Foundation Project of CSTC (cstc2016jcyjA0065) and the NSF of Jiangsu Province (No. BK20161265). The first author was also supported by a grant from the Simons Foundation (No. 499532).

References

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