Powering-invariant normal subgroup of nilpotent group

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This article describes a property that arises as the conjunction of a subgroup property: powering-invariant normal subgroup with a group property imposed on the ambient group: nilpotent group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
This article describes a property that arises as the conjunction of a subgroup property: quotient-powering-invariant subgroup with a group property imposed on the ambient group: nilpotent group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

A subgroup H of a group G is termed a powering-invariant normal subgroup of nilpotent group if it satisfies the following equivalent conditions:

  1. G is a nilpotent group and H is a powering-invariant normal subgroup of G, i.e., H is a normal subgroup and is a powering-invariant subgroup of G. Here, by powering-invariant, we mean if p is a prime number such that G is p-powered, H is also p-powered.
  2. G is a nilpotent group and H is a quotient-powering-invariant subgroup of G, i.e., H is a normal subgroup and if p is a prime number such that G is p-powered, the quotient group G/H is also p-powered.

Equivalence of definitions