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

Abstract: Let $G$ be a finite group. Denoting by ${\rm{cd}}(G)$ 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 ${\rm{cd}}(G)$, and two distinct vertices $p$, $q$ are adjacent if and only if $pq$ divides some number in ${\rm{cd}}(G)$. This paper 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\cong{\rm{PSL}}_2(2^a)) for some integer $a\geq 2$; the remaining case is addressed in the present paper.