Some character degree sets imply direct product

摘要

Let cd(G) be the set of all irreducible complex characters of a finite group G. In [K. Aziziheris, Determining group structure from sets of irreducible character degrees, J. Algebra 323 (2010) 1765-1782], we proved that if m and n are relatively prime integers greater than 1, p is prime not dividing mn, and G is a solvable group such that cd(G) = {1, p, n, m, pn, pm}, then under some conditions on p, m, and n, the group G = A x B is the direct product of two normal subgroups, where cd(A) = {1, p} and cd(B) = {1, n, m}. In this paper, we replace p by u, where u > 1 is an arbitrary positive integer, and we obtain similar result. As an application, we show that if G is a finite group with cd(G) = {1, 21, 13, 55, 273, 1155} or cd(G) = {1, 15, 391, 58, 5865, 870}, then G is a direct product of two non-abelian normal subgroups.