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Template:Function metaproperty satisfaction

Contents 1 Statement 1.1 Verbal statement 1.2 Symbolic statement 2 Proof 2.1 Hands-on proof

Statement

Verbal statement

Any inner automorphism of a subgroup lifts to an inner automorphism of the whole group.

Symbolic statement

Let be groups and σ be an inner automorphism of G. Then, there exists an inner automorphism σ' of H such that the restriction of σ' to G is σ.

Proof

Hands-on proof

We are given and an inner automorphism σ of G. Since σ is an inner automorphism of G, there exists such that σ(x) = gxg ^{− 1}.

Now consider the inner automorphism of H defined via conjugation by g, that is, the map over H. Clearly, this is an inner automorphism of H, and its restriction to G is the same map σ.

This proves the result.