Simplification problem for finite groups

Let $G_1,G_2,H$ be finite groups. My question is: if $G_1\times H$ is isomorphic to $G_2\times H$, is $G_1$ isomorphic to $G_2$?

I came to this question while preparing an exercise on finite abelian groups; note that in the abelian case, the answer is easily seen to be 'yes' using the structure theorem.

I called this the 'simplification problem' in the title of this post by analogy with Zariski' simplification problem for algebraic varieties: if $X\times\mathbb{A}^1$ is isomorphic to $Y\times\mathbb{A}^1$, is $X$ isomorphic to $Y$? However, I don't know anything about the statuts/existence/relevance of the above question among group theorists.

• Commented Dec 7, 2013 at 13:21
• Thanks! So my question is a duplicate. I didn't find the answer after a quick search: I was missing the keyword 'cancellation'. Commented Dec 7, 2013 at 13:44
• I see only four google hits for "Zariski simplification," three of which are the same paper and one of which is this post, but 474 hits for "Zariski cancellation." Commented Dec 8, 2013 at 1:03

Yes (Hirshon), even if $G_1, G_2$ are not finite.