Commutator of a normal subgroup and a subset

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

A subgroup $H$ of a group $G$ is termed a commutator of a normal subgroup and a subset if there exists a normal subgroup $N$ of $G$ and a subset $S$ of $G$ such that $H$ is the commutator $[N,S]$, i.e., the subgroup generated by all the commutators between elements of $N$ and elements of $S$.

Relation with other properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
commutator of a transitively normal subgroup and a subset

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
2-subnormal subgroup	normal subgroup of normal subgroup	commutator of a normal subgroup and a subset implies 2-subnormal		|FULL LIST, MORE INFO