# automophism proof

• Mar 29th 2013, 06:21 PM
henderson7878
automophism proof
Let H and K be groups with relatively prime orders. Show that Aut(HxK)
Aut(H)xAut(K).
• Mar 29th 2013, 07:14 PM
Gusbob
Re: automophism proof
Quote:

Originally Posted by henderson7878
Let H and K be groups with relatively prime orders. Show that Aut(HxK)
Aut(H)xAut(K).

Given $\displaystyle \varphi \in \text{Aut}(H)$ and $\displaystyle \psi \in \text{Aut}(K)$, define a map $\displaystyle \text{Aut}(H) \times \text{Aut}(K) \to \text{Aut}(H\times K)$ by $\displaystyle f: (\varphi,\psi)(h,k)\mapsto (\varphi(h),\varphi(k))$. Show that this is a well defined homomorphism first, then show that it is bijective.
• Mar 30th 2013, 07:21 AM
johng
Re: automophism proof
Hi,
This really isn't very hard, but it is kind of messy. The previous answer is a little misleading in that it looks as though Aut(H) cross Aut(K) would always be isomorphic to Aut(H cross K), which of course is false. Here's the mess:

Attachment 27738

Attachment 27739