Groupg of Automorphisms Aut(G)

1. Jul 22, 2011

arthurhenry

Wikipedia states:
when G splits as direct sum of H and K, then

Aut(H \oplus K) \cong Aut(H) \oplus Aut(K)

Thank you

2. Jul 22, 2011

micromass

Hi arthurhenry!

Where exactly did you see this, what was the context. In general this is false:

$$Aut(\mathbb{Z}_p^2)=GL_2(\mathbb{Z}_p)$$

but

$$Aut(\mathbb{Z}_p)\times Aut(\mathbb{Z}_p)=\mathbb{Z}_{p-1}^2$$

3. Jul 22, 2011

arthurhenry

4. Jul 22, 2011

micromass

OK, next time, could you please state these things completely?? The wikipedia article says that $Aut(H\times K)\cong Aut(H)\times Aut(K)$ if the groups are finite, abelian and of coprime order!! You need those conditions.

As for the proof, try to prove it in these steps

• Given an automorphism $f\times K\rightarrow H\times K$, then f(H)=H and f(K)=K.
• We have a homomorphism

$$\Phi:Aut(H\times K)\rightarrow Aut(H)\times Aut(K):f\rightarrow (f\vert_H, f\vert_K)$$
• Find an inverse of the homomorphism.

5. Jul 22, 2011

arthurhenry

I am sorry, I realized right after I sent the email; and thank you, now I will work on it.

6. Jul 23, 2011

arthurhenry

This might be bad, but I have had problem finding an inverse. I am afraid I am not suing all of the hypothesis either.