# Trivial subgroup is characteristic

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) satisfying a subgroup metaproperty (i.e., trivially true subgroup property)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about characteristic subgroup |Get facts that use property satisfaction of characteristic subgroup | Get facts that use property satisfaction of characteristic subgroup|Get more facts about trivially true subgroup property

## Statement

In any group, the trivial subgroup is a characteristic subgroup, i.e., any automorphism of the whole group maps the trivial subgroup to itself.

## Proof

By definition, a homomorphism of groups must send the identity element to the identity element. Thus, any automorphism of a group must send its identity element to its identity element, and hence, must map the trivial subgroup to within itself.