Closed normal subgroup

From Groupprops
Jump to: navigation, search
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: