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2007-04-01T06:27:46Z

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{{function metaproperty satisfaction}}

==Statement==

===Verbal statement===

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

===Symbolic statement===

Let <math>G \le H</math> be groups and <math>\sigma</math> be an inner automorphism of <math>G</math>. Then, there exists an inner automorphism <math>\sigma'</math> of <math>H</math> such that the restriction of <math>\sigma'</math> to <math>G</math> is <math>\sigma</math>.

==Proof==

===Hands-on proof===

We are given <math>G \le H</math> and an inner automorphism <math>\sigma</math> of <math>G</math>. Since <math>\sigma</math> is an inner automorphism of <math>G</math>, there exists <math>g \in G</math> such that <math>\sigma(x) = gxg^{-1}</math>.

Now consider the inner automorphism of <math>H</math> defined via conjugation by <math>g</math>, that is, the map <math>\sigma' = x \mapsto gxg^{-1}</math> over <math>H</math>. Clearly, this is an inner automorphism of <math>H</math>, and its restriction to <math>G</math> is the same map <math>\sigma</math>.

This proves the result.

Inner is extensibility-stable - Revision history

Vipul: 1 revision

Vipul at 06:27, 1 April 2007