Are the two groups isomorphic?

Question

Let $G,H,K$ be three groups. If $G\times H$ is isomorphic to $G\times K$, then is it true for $H\cong K$?

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score 2 · Accepted Answer · edited May 15 '18 at 09:31

2

No, take

$$G=\mathbb{Z}\times\mathbb{Z}\times\cdots$$ $$H=\mathbb{Z}$$ $$K=1$$

Note that the thesis holds if $G$ is finite (Hirshon's theorem). See here.

edited May 15 '18 at 09:31

Dietrich Burde

130,978

answered May 08 '18 at 08:26

freakish

42,851

Out of curiosity, do you have a counterexample where all 3 groups are finite? – Aaron May 08 '18 at 08:27
2

@Aaron No, the thesis holds for finite groups. – freakish May 08 '18 at 08:27
Thank you for the link, that's quite interesting. – Aaron May 08 '18 at 08:32
Thank you, I understood. – wangkun May 08 '18 at 08:37

score -1 · Answer 2 · answered May 08 '18 at 08:32

-1

Let $A=\prod\limits_{n=1}^{\infty}G_n$, where $G_n=\mathbb{Z}$ for all $n$. And then we have $\mathbb{Z}\times A\cong\mathbb{Z}\times\mathbb{Z}\times A$. But $\mathbb{Z}\ncong\mathbb{Z}\times\mathbb{Z}$.

answered May 08 '18 at 08:32

liwolf

335

Are the two groups isomorphic?

2 Answers2