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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.
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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.
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.
The aim of this paper is to classify the finite nonsolvable groups in which every irreducible character of even degree vanishes on at most two conjugacy classes. As a corollary, it is shown that $L_2(2^f)$ are the only nonsolvable groups in which every irreducible character of even degree vanishes on just one conjugacy class.
To classify the finite dimensional pointed Hopf algebras with $G= {rm HS}$ or ${rm Co3}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {rm GAP}.
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