# Every group is characteristic in itself

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

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

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## Contents

## Statement

Suppose is a group. Then, , viewed as a subgroup of itself, is a characteristic subgroup, i.e., any automorphism of sends to within itself.

## Related facts

### Opposite facts

Note that it is possible for a group to have a subgroup isomorphic to itself that is not characteristic in it. Explicitly, consider a group that is an countable direct power of a nontrivial group. This group is not characteristic as a subgroup of its direct product with itself, even though it is isomorphic to that direct product.

## Proof

The proof follows by definition: any automorphism of a group must send it to within itself.