On Finite Quasi-Core-p p-Groups

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.