Planar character-graphs

Abstract

For a finite group G, let R(G) be the solvable radical of G. The character-graph $\Delta (G)$ of G is a graph whose vertices are the primes which divide the degrees of some irreducible complex characters of G and two distinct primes p and q are joined by an edge if the product pq divides some character degree of G. In this paper we prove that, if $\Delta (G)$ has no subgraph isomorphic to $K_{3,3}$ and it’s complement is non-bipartite, then $\frac{G}{R(G)}$ is an almost simple group with socle isomorphic to ${\text{PSL}}_2(q)$ where $q⩾5$ is a prime power. Also we study the structure of all planar graphs that occur as the character-graph $\Delta (G)$ of a finite group G.

Keywords:

• Character degree
• character-graph
• planar graph

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20C15
• Secondary: 05C10
• 05C25

Additional information

Funding

This research was supported in part by a grant from School of Mathematics, Institute for Research in Fundamental Sciences (IPM).