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

Let G be a finite group. Denoting by cd (G) the set of the 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). This paper completes the classification, started in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119), of the finite non-solvable groups whose character degree graph has a cut-vertex, i.e., a vertex whose removal increases the number of connected components of the graph. More specifically, it was proved in Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119 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 Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. II, 2022. https://doi.org/10.1007/s10231-022-01299-3) and Dolfi et al. (Non-solvable groups whose character degree graph has a cut-vertex. I, 2022. https://doi.org/10.48550/arXiv.2207.10119) all isomorphism types for S were treated, except the case S≅ PSL 2(2 a) for some integer a≥ 2 ; the remaining case is addressed in the present paper.