# Normal subgroup

• October 3rd 2011, 06:43 AM
kierkegaard
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  $\phi$ of H', there is an element $g\inG$ such that
$\phi = I_{g}$ on H', where $I_{g}$ is the inner automorphism associated to g.

Thanks.
• October 3rd 2011, 08:23 AM
Swlabr
Re: Normal subgroup
Quote:

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 $\phi$ of H', there is an element $g\inG$ such that
$\phi = I_{g}$ on H', where $I_{g}$ 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?...