Cancellation of Groups in Internal Direct Products

[Oster]

G, H, K are groups. G is finite. GxH is isomorphic to GxK. Prove H is isomorphic to K. Give an example to show that this does not hold when G is infinite.

The counter example when G is infinite is Rx{0} and RxR (R - real numbers)
I'm having trouble Proving the main part of the question. I have a hunch that the image of Gx{0} in GxK will be the direct product of a subgroup of G and a subgroup of K. Can someone help me?

 

[pasmith]

G, H, K are groups. G is finite. GxH is isomorphic to GxK. Prove H is isomorphic to K.

The first observation is that, for an isomorphism to exist between finite groups, the orders of the groups must be equal. So immediately H and K are of the same order.

Any isomorphism $\phi : G \times H \to G \times K$ can be written in terms of maps $\theta: G \to G$ and $\psi : H \to K$ as
$$\phi(g,h) = (\theta(g),\psi(h))$$
Now work out the conditions $\theta$ and $\psi$ must satisfy for $\phi$ to be an isomorphism.

 

[Oster]

H and K need not be finite. And I don't see why your second claim should hold. Can you explain a bit more please?

 

[pasmith]

H and K need not be finite. And I don't see why your second claim should hold. Can you explain a bit more please?

I should have $\phi(g,h) = (\theta(g,h),\psi(g,h))$ for $\theta : G \times H \to G$ and $\psi : G \times H \to K$. The idea is then to look at $\phi(1,h)$.