# Math Help - automophism proof

1. ## automophism proof

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

2. ## Re: automophism proof

Originally Posted by henderson7878
Let H and K be groups with relatively prime orders. Show that Aut(HxK)
Aut(H)xAut(K).
Given $\varphi \in \text{Aut}(H)$ and $\psi \in \text{Aut}(K)$, define a map $\text{Aut}(H) \times \text{Aut}(K) \to \text{Aut}(H\times K)$ by $f: (\varphi,\psi)(h,k)\mapsto (\varphi(h),\varphi(k))$. Show that this is a well defined homomorphism first, then show that it is bijective.

3. ## 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: