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.
Highlights
Let G be a group and H is a subgroup of G
G has an abelian normal subgroup of index bounded in terms of n only
In [2], Lennox, Smith and Wiegold show that, for p 6= 2, a core-p p-group is nilpotent of class at most 3 and has an abelian normal subgroup of index at most p5
Summary
Let G be a group and H is a subgroup of G. Lennox, Neumaan, Smith and Wiegold investigated the core-n groups in [1]. They show that every locally finite group. In [2], Lennox, Smith and Wiegold show that, for p 6= 2, a core-p p-group is nilpotent of class at most 3 and has an abelian normal subgroup of index at most p5. Cutolo, Khukhro, Lennox, Wiegold, Rinauro and Smith [3] prove that a core-p p-group G has a normal abelian subgroup whose index in G is at most p2 if p 6= 2. If p = 2, Cutolo, Smith and Wiegold [4] prove that every core-2 2-group has an abelian subgroup of index at most 16. We call this kind of p-groups quasi-core-p p-groups
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