The sub-class sizes of some elements being square free

Abstract

Let x be an element of a finite group G, and p a prime factor of the order of G. It is clear that there always exists a unique minimal subnormal subgroup containing x, say [Formula: see text]. We call the conjugacy class of x in [Formula: see text] the sub-class of x in G, see [G. Qian and Y. Yang, On sub-class sizes of finite groups, J. Aust. Math. Soc. (2020) 402–411]. In this paper, assume that [Formula: see text] is the product of the subgroups A and B, we investigate the solvability, p-nilpotence and supersolvability of the group G under the condition that the sub-class sizes of prime power order elements in [Formula: see text] are [Formula: see text] free, [Formula: see text] free and square free, respectively, so that some known results relevant to conjugacy class sizes are generalized.