# Normal closure of finite subset

From Groupprops

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]

## Contents

## Definition

A subgroup of a group is termed a **normal closure of finite subset** if there is a finite subset of that subgroup such that the normal subgroup generated by that finite subset in the *whole group* is the subgroup. In other words, the subgroup arises as the normal closure in the whole group of a finitely generated subgroup.