The character co-degree sets and solvability

Abstract

For a character [Formula: see text] of a finite group [Formula: see text], the number [Formula: see text] is called the co-degree of [Formula: see text]. In this paper, we concentrate on some conditions on the character co-degree sets implying solvability. More precisely, we show that if for every non-principal irreducible characters [Formula: see text] and [Formula: see text] of [Formula: see text] with [Formula: see text], the greatest common divisor of [Formula: see text] and [Formula: see text] is divisible by at most two primes (counting multiplicities), then either [Formula: see text] is solvable or [Formula: see text] is isomorphic to one of the groups [Formula: see text], [Formula: see text] or an extension of an elementary abelian [Formula: see text]-group [Formula: see text] of order [Formula: see text] by [Formula: see text]. In the last case, the set of the co-degrees of the irreducible characters of [Formula: see text] is [Formula: see text].