Structure of Groups with At Most Six Character Degrees

Abstract

Let a, b, c and d be relatively prime integers greater than 1 and cd(G) be the set of all irreducible complex character degrees of a finite group G. Mark Lewis in [4] proved that every group G with  cd(G) = {1,  p, q, r, pq, pr},where p, q and r are distinct primes, is the direct product of two groups H and K with cd(H)={1, p, q} ; cd(K)={1, p, r} and so is solvable. We try to use classification of finite groups and drop the primness hypothesis and show that the same result holds in a special case. In fact we show that if a, b, c and d be relatively prime integers greater than 1 and cd(G) 14⊆"> {1, a, bd, cd, abd, acd}, then G is solvable. Also as an interesting application of this theorem we prove that there is no group G with cd(G) = {1, a,  bd,  cd}, where a, b, c and d be relatively prime integers greater than 1.

Keywords: Character degree, Solvable group, Irreducible character degree set, Degree graph. 2010 AMS Mathematics Subject Classification: 20C15; 20D05.