Thread: Normal subgroup

Let H be a group. Prove there is a group G containing a normal subgroup H' with the following two properties:

(a) The group H is isomorphic to H' , and
(b) For every automorphism  of H', there is an element such that 
on H', where is the inner automorphism associated to g.

Thanks.

Originally Posted by kierkegaard

Let H be a group. Prove there is a group G containing a normal subgroup H' with the following two properties:

(a) The group H is isomorphic to H' , and
(b) For every automorphism of H', there is an element such that
on H', where is the inner automorphism associated to g.

Thanks.

Can you not just stack some semi-direct products up?...(so the induced automorphism is trivial on the extended bit, and induces the automorphism on H').

EDIT: Alternatively, although this might be overkill, do you know what an HNN-extension is?...