Character degree graphs that are complete graphs

Bianchi, Mariagrazia; Chillag, David; Lewis, Mark; Pacifici, Emanuele

doi:10.1090/S0002-9939-06-08651-5

Character degree graphs that are complete graphs
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by Mariagrazia Bianchi, David Chillag, Mark L. Lewis and Emanuele Pacifici PDF

Proc. Amer. Math. Soc. 135 (2007), 671-676 Request permission

Abstract:

Let $G$ be a finite group, and write $\operatorname {cd}(G)$ for the set of degrees of irreducible characters of $G$. We define $\Gamma (G)$ to be the graph whose vertex set is $\operatorname {cd}(G)-\{1\}$, and there is an edge between $a$ and $b$ if $(a,b)>1$. We prove that if $\Gamma (G)$ is a complete graph, then $G$ is a solvable group.

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