On Finite Quasi-Core-<i>p p</i>-Groups

Abstract

Given a positive integer n, a finite group G is called quasi-core-n if 〈 x 〉 / 〈 x 〉 G has order at most n for any element x in G, where 〈 x 〉 G is the normal core of 〈 x 〉 in G. In this paper, we investigate the structure of finite quasi-core-p p-groups. We prove that if the nilpotency class of a quasi-core-p p-group is p + m , then the exponent of its commutator subgroup cannot exceed p m + 1 , where p is an odd prime and m is non-negative. If p = 3 , we prove that every quasi-core-3 3-group has nilpotency class at most 5 and its commutator subgroup is of exponent at most 9. We also show that the Frattini subgroup of a quasi-core-2 2-group is abelian.

Keywords

• finite <i>p</i>-group
• quasi-core-<i>p p</i>-group
• commutator subgroup