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Register a new userOn p-parts of character degrees and conjugacy class sizes of finite groups
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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.
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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).
In this paper, we get the sharp bound for $|G/O_p(G)|_p$ under the assumption that either $p^2 mid chi(1)$ for all $chi in {rm Irr}(G)$ or $p^2 mid phi(1)$ for all $phi in {rm IBr}_p(G)$. This would settle two conjectures raised by Lewis, Navarro, Tiep, and Tong-Viet.
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.
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.
We show that the largest character degree of an alternating group $A_n$ with $ngeq 5$ can be bounded in terms of smaller degrees in the sense that [ b(A_n)^2<sum_{psiintextrm{Irr}(A_n),,psi(1)< b(A_n)}psi(1)^2, ] where $textrm{Irr}(A_n)$ and $b(A_n)$ respectively denote the set of irreducible complex characters of $A_n$ and the largest degree of a character in $textrm{Irr}(A_n)$. This confirms a prediction of I. M. Isaacs for the alternating groups and answers a question of M. Larsen, G. Malle, and P. H. Tiep.
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