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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.

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    $\begingroup$ See mathoverflow.net/questions/83395/… $\endgroup$
    – S. Carnahan ♦
    Commented Dec 7, 2013 at 13:21
  • $\begingroup$ Thanks! So my question is a duplicate. I didn't find the answer after a quick search: I was missing the keyword 'cancellation'. $\endgroup$
    – Matthieu Romagny
    Commented Dec 7, 2013 at 13:44
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    $\begingroup$ 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." $\endgroup$
    – Qiaochu Yuan
    Commented Dec 8, 2013 at 1:03

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

Related: Can we ascertain that there exist an epimorphism $G\rightarrow H?$

Cancellation Theorem for groups

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