# 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]

## 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