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

2022/09/15 by Silvio Dolfi, Emanuele Pacifici, Dolfi, S. +3
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2209.07161

openalex publication_date 2022/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a finite group. Denoting by \rmcd(G) the set of the degrees of the irreducible complex characters of G, we consider the \it character degree graph of G: this is the (simple, undirected) graph whose vertices are the prime divisors of the numbers in \rmcd(G), and two distinct vertices p, q are adjacent if and only if pq divides some number in \rmcd(G). This paper completes the classification, started in [5] and [6], of the finite non-solvable groups whose character degree graph has a \it cut-vertex, i.e. a vertex whose removal increases the number of connected components of the graph. More specifically, it was proved in [6] that these groups have a unique non-solvable composition factor S, and that S is isomorphic to a group belonging to a restricted list of non-abelian simple groups. In [5] and [6] all isomorphism types for S were treated, except the case \(S≅\rmPSL₂(2^a)\) for some integer a≥ 2; the remaining case is addressed in the present paper.

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