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ON $p$-PARTS OF CONJUGACY CLASS SIZES OF FINITE GROUPS
Published online by Cambridge University Press: 28 March 2018
Abstract
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 $|G:F(G)|_{p}\leq p^{2}$.
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- © 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).
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