Non-divisibility among character degrees

Abstract

In this paper, we study groups for which if 1 < a < b are character degrees, then a does not divide b. We say that these groups have the condition no divisibility among degrees (NDAD). We conjecture that the number of character degrees of a group that satisfies NDAD is bounded and we prove this for solvable groups. More precisely, we prove that solvable groups with NDAD have at most four character degrees and have derived length at most 3. We give a group-theoretic characterization of the solvable groups satisfying NDAD with four character degrees. Since the structure of groups with at most three character degrees is known, these results describe the structure of solvable groups with NDAD.