# Closed normal subgroup

From Groupprops

This article defines a property that can be evaluated for a subgroup of a semitopological group

## Definition

A **closed normal subgroup** of a topological group is a subgroup that is both a closed subgroup (i.e., it is closed as a subset of the whole group with the underlying topology) and a normal subgroup.

### For a Lie group

When the term **closed normal subgroup** is used in the context of a Lie group, we mean it with respect to the topology induced by the Lie structure.

### For an algebraic group

When the term **closed normal subgroup** is used in the context of an algebraic group, we mean it with respect to the Zariski topology on the underlying algebraic variety.

## Facts

When a group has both an algebraic group and a Lie group structure, the notion of closed normal subgroup with respect to the algebraic group structure may be strictly stronger than that with respect to the Lie group structure: