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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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Given a group $G$ and a subgroup $H$, we let $mathcal{O}_G(H)$ denote the lattice of subgroups of $G$ containing $H$. This paper provides a classification of the subgroups $H$ of $G$ such that $mathcal{O}_{G}(H)$ is Boolean of rank at least $3$, when $G$ is a finite alternating or symmetric group. Besides some sporadic examples and some twist
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.
Let $p$ be a fixed prime. For a finite group generated by elements of order $p$, the $p$-width is defined to be the minimal $kinmathbb{N}$ such that any group element can be written as a product of at most $k$ elements of order $p$. Let $A_{n}$ denote the alternating group of even permutations on $n$ letters. We show that the $p$-width of $A_{n}$ $(ngeq p)$ is at most $3$. This result is sharp, as there are families of alternating groups with $p$-width precisely 3, for each prime $p$.
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.
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