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.