Some character degree sets imply direct product

Abstract

Let [Formula: see text] be the set of all irreducible complex characters of a finite group [Formula: see text]. In [K. Aziziheris, Determining group structure from sets of irreducible character degrees, J. Algebra 323 (2010) 1765–1782], we proved that if [Formula: see text] and [Formula: see text] are relatively prime integers greater than [Formula: see text], [Formula: see text] is prime not dividing [Formula: see text], and [Formula: see text] is a solvable group such that [Formula: see text], then under some conditions on [Formula: see text] and [Formula: see text], the group [Formula: see text] is the direct product of two normal subgroups, where [Formula: see text] and [Formula: see text]. In this paper, we replace [Formula: see text] by [Formula: see text], where [Formula: see text] is an arbitrary positive integer, and we obtain similar result. As an application, we show that if [Formula: see text] is a finite group with [Formula: see text] or [Formula: see text], then [Formula: see text] is a direct product of two non-abelian normal subgroups.