Powering-invariant normal subgroup of nilpotent group

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

• (1) implies (2) follows from normal subgroup of nilpotent group satisfies the subgroup-to-quotient powering-invariance implication.
• (2) implies (1) follows from quotient-powering-invariant implies powering-invariant.