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Automorphisms and Conjugation

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Show that if H is any group then there is a group G that contains H as a normal subgroup with the property that for every automorphism f of H there is an element g of G such that the conjugation by g when restricted to H is the given automorphism f, i.e every automorphism of H obtained as an inner automorphism of G restricted to H.

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Solution Summary

Automorphisms and Conjugation are investigated.

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Proof:
We have an identity map id from Aut(H) to itself, such that id(f)=f
for any f in Aut(H). We define G=H x_id Aut(H).
Then for any (h1,f1), (h2,f2) in G, we have
(h1,f1)*(h2,f2)=(h1*id(f1)(h2),f1f2)=(h1f1(h2),f1f2).
H and Aut(H) can be ...

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