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Are there four non-trivial groups $G_1,G_2,H_1,H_2$ pairwise non-isomorphic but such that $G_1 \times G_2$ is isomorphic to $H_1 \times H_2$?

My idea was to consider the abelian groups and something related to their decomposition as direct product of cyclic groups, but it doesn't work.

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    $\begingroup$ Try again with abelian groups of the form $C_2^k = C_2 \times C_2 \times \cdots \times C_2$ with $k$ direct factors. It's easy. $\endgroup$ – Derek Holt May 16 '17 at 8:20
  • $\begingroup$ What do you mean for $C_2^k$? $\endgroup$ – TheWanderer May 16 '17 at 8:21
  • $\begingroup$ $C_2$ meaning $Z/2Z$. Can use small finite products. $\endgroup$ – quasi May 16 '17 at 8:23
  • $\begingroup$ I think the minimal $\;k\;$ for this to fully work is $\;k=5\;$ ... $\endgroup$ – DonAntonio May 16 '17 at 8:34
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    $\begingroup$ @TheWanderer: Hint: 2 + 3 = 1 + 4 $\endgroup$ – quasi May 16 '17 at 8:35
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Let $C_n$ denote the cyclic group with $n$ elements.

Relevant fact: If $a$ and $b$ are relatively prime, then $C_a \times C_b = C_{ab}$.

Idea of proof of fact (or look in any group theory book): It suffices to show that $C_{ab}$ is cyclic - i.e. is generated by a single element. The naive guess is $(1,1)$. Observe that $a*(1,1) = (0,a)$, and since $a$ is a unit modulo $b$ (this is equivalent to them being relatively prime), there is some c so that $ca*(1,1) = (0,1)$. Similarly you can get $(1,0)$ as a mutliple of $(1,1)$.

Do you see how to use this fact?

There are other ways also, such as what is being suggested in the comments.

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It would be more fun if we were to insist that $G_{1}\simeq G_{2}$ and $H_{1}\simeq H_2$ and then give a counterexample.

Tony Corner's example of an abelian group $A\not\simeq A\oplus A$ but with $A\simeq A\oplus A\oplus A$ will give an example. Take $B=A\oplus A$ and then $$ A\oplus A\simeq A\oplus (A\oplus A\oplus A)\simeq (A\oplus A)\oplus (A\oplus A)\simeq B\oplus B $$

(For the group $A$ see MR0169905 Corner, A.L.S., On a conjecture of Pierce concerning direct decomposition of Abelian groups. 1964 Proc. Colloq. Abelian Groups (Tihany, 1963) pp.43--48 Akademiai Kiado, Budapest.)

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