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Groupprops β

Every group is characteristic in itself

Statement

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