# Topological automorphism-invariant subgroup

From Groupprops

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

ANALOGY: This is an analogue in topological group of a property encountered in group. Specifically, it is a topological subgroup property analogous to the subgroup property: characteristic subgroup

View other analogues of characteristic subgroup | View other analogues in topological groups of subgroup properties (OR, View as a tabulated list)

## Contents

## Definition

### Symbol-free definition

A subgroup of a topological group is said to be **topological automorphism-invariant** or **topologically characteristic** if it is invariant under all the topological automorphisms of the topological group.

### Definition with symbols

A subgroup of a topological group is said to be **topological automorphism-invariant** or **topologically characteristic** in if given a topological automorphism ( is thus both a homeomorphism and an automorphism), .

## Importance

Any topological subgroup-defining function on topological groups, must always output a topologically characteristic subgroup.