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Let $F(G)$ and $b(G)$ respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group $G$. A well-known conjecture of D. Gluck claims that if $G$ is solvable then $|G:F(G)|leq b(G)^{2}$. We confirm this conjecture in the case where $|F(G)|$ is coprime to 6. We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure.
This paper presents the best known bounds for a conjecture of Gluck and a conjecture of Navarro.
Building on earlier papers of several authors, we establish that there exists a universal constant $c > 0$ such that the minimal base size $b(G)$ of a primitive permutation group $G$ of degree $n$ satisfies $log |G| / log n leq b(G) < 45 (log |G| / l
We prove a conjecture made by Gilman in 1984 that the groups presented by finite, monadic, confluent rewriting systems are precisely the free products of free and finite groups.
The non--commuting graph $Gamma(G)$ of a non--abelian group $G$ is defined as follows. The vertex set $V(Gamma(G))$ of $Gamma(G)$ is $Gsetminus Z(G)$ where $Z(G)$ denotes the center of $G$ and two vertices $x$ and $y$ are adjacent if and only if $xy
We prove Wises $W$-cycles conjecture. Consider a compact graph $Gamma$ immering into another graph $Gamma$. For any immersed cycle $Lambda:S^1to Gamma$, we consider the map $Lambda$ from the circular components $mathbb{S}$ of the pullback to $Gamma$.