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I'm looking for an example of $A,B,C$ such that $A\times B \cong C\times B$ but $A,C$ are not isomorphic.

I've tried many infinite groups but none get to the answer,any hint would be appreciated.

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  • $\begingroup$ I'm not sure why is the question closed,I can't see any unclear point in it! $\endgroup$
    – MAh2014
    Mar 1, 2017 at 18:21

1 Answer 1

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Let $A={\mathbb Z}_2$, $C = {\mathbb Z}_2\times {\mathbb Z}_2$ and $B={\mathbb Z}_2\times{\mathbb Z}_2 \times {\mathbb Z}_2\times \dots $.

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  • $\begingroup$ For OP: this generalizes extremely easily to any group. $\endgroup$ Feb 22, 2017 at 21:57
  • $\begingroup$ The example also shows that $A$ and $C$ do not have to be isomorphic: if you see $A \times B \cong C \times B$ then you are inclined to say, hey divide out at both sides by $\{1\} \times B$, so $A$ and $C$ must be isomorphic. Unfortunately, not true! $\endgroup$ Feb 22, 2017 at 21:59
  • $\begingroup$ Is $B$ infinite direct product of ${\mathbb Z}_2 $?How is this a group?,and sorry I ment $A,C$ are not isomorphic ,just edited. $\endgroup$
    – MAh2014
    Feb 22, 2017 at 22:09
  • $\begingroup$ Now I am puzzled. An infinite direct product? Of course this is a group! And @Cameron Williams: "to any finite group". $\endgroup$ Feb 22, 2017 at 22:24
  • $\begingroup$ Sorry if I asked an prime question,I just learnt about the topic.and Thanks for helping me out. $\endgroup$
    – MAh2014
    Feb 22, 2017 at 22:31

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