# Every injective endomorphism arises as the restriction of an inner automorphism

From Groupprops

## Statement

Suppose is a group and is an Injective endomorphism (?) of . Then, there exists a group containing as a subgroup, as well as an Inner automorphism (?) of , such that the restriction of to equals .

Notice that any endomorphism that arises as the restriction of an inner automorphism *must* be injective, hence this result is tight.

## Related facts

- Inner automorphism to automorphism is right tight for normality: Every automorphism arises as the restriction of an inner automorphism from a bigger group in which the given group is normal.
- Left transiter of normal is characteristic
- Characteristic of normal implies normal
- Isomorphic iff potentially conjugate: Any isomorphism between two subgroups of a group can be realized as the restriction of an inner automorphism in some bigger group. This uses the natural construction of HNN-extensions.