Finite nonsolvable groups with many distinct character degrees

Hung P. Tong-Viet

Vol. 268 (2014), No. 2, 477–492

DOI: 10.2140/pjm.2014.268.477

Abstract

Let $G$ be a finite group and let $Irr\left(G\right)$ denote the set of all complex irreducible characters of $G$. Let $cd\left(G\right)$ be the set of all character degrees of $G$. For a degree $d\in cd\left(G\right)$, the multiplicity of $d$ in $G$, denoted by $m_G\left(d\right)$, is the number of irreducible characters of $G$ having degree $d$. A finite group $G$ is said to be a $T_k$-group for some integer $k\ge 1$ if there exists a nontrivial degree $d_0\in cd\left(G\right)$ such that $m_G\left(d_0\right)=k$ and that for every $d\in cd\left(G\right)-\left\{1,d_0\right\}$, the multiplicity of $d$ in $G$ is trivial, that is, $m_G\left(d\right)=1$. In this paper, we show that if $G$ is a nonsolvable $T_k$-group for some integer $k\ge 1$, then $k=2$ and $G\cong {PSL}_2\left(5\right)$ or ${PSL}_2\left(7\right)$.

Keywords

multiplicity, character degrees, nonsolvable groups

Mathematical Subject Classification 2010

Primary: 20C15

Secondary: 20C33, 20D05

Milestones

Received: 25 October 2012

Revised: 5 December 2012

Accepted: 10 December 2012

Published: 21 June 2014