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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.
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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.
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.
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.
Let $G$ be a finite group and $mathrm{Irr}(G)$ be the set of irreducible characters of $G$. The codegree of an irreducible character $chi$ of the group $G$ is defined as $mathrm{cod}(chi)=|G:mathrm{ker}(chi)|/chi(1)$. In this paper, we study two topics related to the character codegrees. Let $sigma^c(G)$ be the maximal integer $m$ such that there is a member in $mathrm{cod}(G)$ having $m$ distinct prime divisors, where $mathrm{cod}(G)={mathrm{cod}(chi)|chiin mathrm{Irr}(G)}$. One is related to the codegree version of the Hupperts $rho$-$sigma$ conjecture and we obtain the best possible bound for $|pi(G)|$ under the condition $sigma^c(G) = 2,3,$ and $4$ respectively. The other is related to the prime graph of character codegrees and we show that the codegree prime graphs of several classes of groups can be characterized only by graph theoretical terms.
In this paper we measure how efficiently a finite simple group $G$ is generated by its elements of order $p$, where $p$ is a fixed prime. This measure, known as the $p$-width of $G$, is the minimal $kin mathbb{N}$ such that any $gin G$ can be written as a product of at most $k$ elements of order $p$. Using primarily character theoretic methods, we sharply bound the $p$-width of some low rank families of Lie type groups, as well as the simple alternating and sporadic groups.
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