Nonsolvable Groups with no Prime Dividing Four Character Degrees

Given a finite group G, we say that G has property if every set of k distinct irreducible character degrees of G is setwise relatively prime. In this paper, we show that if G is a finite nonsolvable group satisfying then G has at most 8 distinct character degrees. Combining with work of D. Benjamin on finite solvable groups, we deduce that a finite group G has at most 9 distinct character degrees if G has property and this bound is sharp.