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.



Simplification problem for finite groups
Yes ([Hirshon](http://www.jstor.org/discover/10.2307/2317133?uid=3737864&uid=2&uid=4&sid=21103081075687])), even if $G\_1, G\_2$ are not finite.


Related: [Can we ascertain that there exist an epimorphism $G\rightarrow H?$](https://mathoverflow.net/questions/114139/)


[Cancellation Theorem for groups](https://mathoverflow.net/questions/83395/)


