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The largest size of conjugacy class and the $p$-parts of finite groups

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 Added by Yong Yang
 Publication date 2017
  fields
and research's language is English




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Let $p$ be a prime and let $P$ be a Sylow $p$-subgroup of a finite nonabelian group $G$. Let $bcl(G)$ be the size of the largest conjugacy class of the group $G$. We show that $|P/O_p(G)| < bcl(G)$ if $G$ is not abelian.



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134 - Yong Yang , Guohua Qian 2015
Let $G$ be a finite group and $Irr(G)$ the set of irreducible complex characters of $G$. Let $e_p(G)$ be the largest integer such that $p^{e_p(G)}$ divides $chi(1)$ for some $chi in Irr(G)$. We show that $|G:mathbf{F}(G)|_p leq p^{k e_p(G)}$ for a constant $k$. This settles a conjecture of A. Moreto. We also study the related problems of the $p$-parts of conjugacy class sizes of finite groups.
It is proved that, for a prime $p>2$ and integer $ngeq 1$, finite $p$-groups of nilpotency class $3$ and having only two conjugacy class sizes $1$ and $p^n$ exist if and only if $n$ is even; moreover, for a given even positive integer, such a group is unique up to isoclinism (in the sense of Philip Hall).
224 - E. A. Ivanova 2004
It is proved that for any prime $p$ a finitely generated nilpotent group is conjugacy separable in the class of finite $p$-groups if and only if the torsion subgroup of it is a finite $p$-group and the quotient group by the torsion subgroup is abelian.
In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is $0$ for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group.
160 - Boris Weisfeiler 2012
This is a nearly complete manuscript left behind by Boris Weisfeiler before his disappearance during a hiking trip in Chile in 1985. It is posted on a request from the authors sister, Olga Weisfeiler.
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