Throughout this note, G will be a finite group, IrrΓ°GΓ will be the set of irreducible characters of G, cdΓ°GΓ will be the set of character degrees of G, and rΓ°GΓ will be the set of primes that divide degrees in cdΓ°GΓ. The prime vertex degree graph of G, written DΓ°GΓ, is the graph with rΓ°GΓ as its vertex set, and with an edge between p and q if pq divides some degree a A cdΓ°GΓ. An overview of the literature on these graphs can be found in .
It seems that most groups G have graphs DΓ°GΓ that are complete graphs (although we do not want to be precise on what we mean by 'most'). For solvable groups, it has been shown that if DΓ°GΓ is not complete, then the structure of G is limited (see and).
In this note, we show that a large class of non-solvable groups have degree graphs that are complete. To study the degree graphs of non-solvable groups, a standard starting point is with the degree graphs of non-abelian simple groups. In a series of papers, Don White has classified the character degree graphs of all non-abelian simple groups. Except for some examples of small rank, nearly all non-abelian simple groups have degree graphs that are complete graphs. These results are summarized in .
Extending this idea, this note looks at the characteristically simple groups, groups for which there are exactly two characteristic subgroups. Each such group is a direct product of copies of a fixed simple group. With this in mind, we let G n denote G Γ Γ Γ Γ Γ G (n copies); thus S n is characteristically simple when S itself is a simple group. We now fix S to be a non-abelian simple group. Because the non-abelian simple groups have already been addressed, we restrict our attention to when n > 1. However, it is relatively easy to show that DΓ°S n Γ is complete for n > 1. More generally, the graph DΓ°H n Γ associated with any non-abelian group H is always complete for every integer n > 1. Thus we expand our scope and consider extending these groups. We recognize first that S n is isomorphic to InnΓ°S n Γ. Identifying S n with its set of inner automorphisms, our goal herein is to prove the following result.