Nonsolvable Groups Satisfying the One-Prime Hypothesis

Throughout this paper, G is a finite group and Irr(G) is the set of irreducible characters of G. We are particularly interested in the values these characters take on the identity of G. If χ ∈ Irr(G), then χ(1) is the degree of χ. The set of all degrees for G is written cd(G) = {χ(1) |χ ∈ Irr(G)}. In recent years, there has been much interest in finding connections between the structure of a finite group G and the structure of the character degree set cd(G). One fruitful approach has been to study properties of certain graphs associated with cd(G). One such graph is ∆(G), whose vertices are the prime divisors of degrees in cd(G), with vertices p and q joined by an edge if pq divides some degree in cd(G). This graph has been studied in [22], [23], [18], [19], and [21], for example. A closely related graph is the graph Γ(G), whose set of vertices is cd(G) itself, with an edge joining two degrees if and only if the degrees have a common prime divisor (see [22], [25], and [26]). These graphs tend to be more interesting when there are few common divisors among degrees. It is also interesting to find other ways to study the situation where there are few common divisors among degrees. It can be shown that if G is a solvable group whose distinct character degrees are relatively prime, then |cd(G)| 6 3 (see [12, Problem 12.3]). The same bound on |cd(G)| was proved by Isaacs and Passman [13] for arbitrary finite groups under the more restrictive hypothesis that every nonlinear character has prime degree. Studies of various weakenings of the coprimeness condition have followed this work. Benjamin [1] and McVey [27] have proved bounds on |cd(G)| in terms of the maximum number of distinct character degrees that are divisible by any prime. Lewis, Moretó, and Wolf [17] and Malle and Moretó [28] have classified groups in which no nontrivial character degree divides another distinct degree. In all of these studies of the relationship between the divisibility structure of cd(G) and the group structure, the solvable cases have been studied more extensively, and it is only recently that the nonsolvable cases have begun to receive attention. In this paper, we consider another way of looking at the divisibility structure of the set of character degrees. In particular, we focus on groups satisfying the one-prime hypothesis. This is another weakening of the coprimeness condition that was introduced by the first author in [15]. We say that G satisfies the one-prime hypothesis if the greatest common divisor of every pair of distinct elements of cd(G) is 1 or a prime. In [15], the first author showed that if G is a solvable group satisfying the one-prime hypothesis, then |cd(G)| ≤ 14. In a more recent paper [16], the first