Bounding the Derived Length for a Given Set of Character Degrees

Abstract

Let G be a finite solvable group with {1, a, b, c, ab, ac} as the character degree set, where a ,b, and c are pairwise coprime integers greater than 1. We show that the derived length of G is at most 4. This verifies that the Taketa inequality, dl(G) ≤ |cd(G)|, is valid for solvable groups with {1, a, b, c, ab, ac} as the character degree set. Also, as a corollary, we conclude that if a, b, c, and d are pairwise coprime integers greater than 1 and G is a solvable group such that cd(G) = {1, a, b, c, d, ac, ad, bc, bd}, then dl(G) ≤ 5. Finally, we construct a family of solvable groups whose derived lengths are 4 and character degree sets are in the form {1, p, b, pb, qp, pqp}, where p is a prime, q is a prime power of an odd prime, and b > 1 is integer such that p, q, and b are pairwise coprime. Hence, the bound 4 is the best bound for the derived length of solvable groups whose character degree set is in the form {1, a, b, c, ab, ac} for some pairwise coprime integers a, b, and c.