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

TL;DR

This paper investigates the structure of non-solvable finite groups based on the properties of their character degree graphs, specifically focusing on cases where the graph has a cut-vertex, which affects the graph's connectivity.

Contribution

It initiates a series analyzing non-solvable groups with character degree graphs having a cut-vertex, providing foundational insights into their structure.

Findings

Character degree graph with a cut-vertex influences group structure.

Non-solvable groups exhibit specific patterns in their character degree graphs.

This work sets the stage for further classification in subsequent papers.

Abstract

Let G be a finite group. Denoting by cd(G) the set of degrees of the irreducible complex characters of G, we consider the character degree graph of G: this is the (simple undirected) graph whose vertices are the prime divisors of the numbers in cd(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in cd(G). In the series of three papers starting with the present one, we analyze the structure of the finite non-solvable groups whose character degree graph possesses a cut-vertex, i.e., a vertex whose removal increases the number of connected components of the graph.