Non-solvable groups whose character degree graph has a cut-vertex. II

TL;DR

This paper classifies certain non-solvable finite groups based on the structure of their character degree graph, specifically those with a cut-vertex, focusing on groups with odd prime power factors greater than 5.

Contribution

It extends previous classification work by analyzing non-solvable groups with specific prime power factors in their character degree graph, completing the case for odd primes greater than 5.

Findings

Classifies non-solvable groups with a cut-vertex in their character degree graph for odd prime powers > 5.

Complements prior work by covering cases with odd prime factors, setting the stage for the final case with t=2.

Provides structural insights into the prime divisors influencing the graph connectivity of group characters.

Abstract

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. Define then the character degree graph $\Delta(G)$ as the (simple undirected) graph whose vertices are the prime divisors of the numbers in ${\rm{cd}}(G)$, and two distinct vertices $p$, $q$ are adjacent if and only if $pq$ divides some number in ${\rm{cd}}(G)$. This paper continues the work, started in [7], toward the classification of the finite non-solvable groups whose degree graph possesses a cut-vertex, i.e., a vertex whose removal increases the number of connected components of the graph. While, in [7], groups with no composition factors isomorphic to ${\rm{PSL}}_2(t^a)$ (for any prime power $t^a\geq 4$) were treated, here we consider the complementary situation in the case when $t$ is odd and $t^a> 5$. The proof of this classification will be then completed in the third and last paper of this series ([8]), that deals with the case $t=2$.