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Let $(V,٭)$ be a group where $V=\{a,b,c,d\}$. $(V,٭)$ has the property that every element is an inverse of itself, so $V$ is called the $V$ group or the "Klein 4 group".

I would like to know whether $(V,٭)$ is isomorphic to $(\mathbb{Z}_4,+_4)$. How may I achieve this? Should I try to show that every element of $(\mathbb{Z}_4,+_4)$ is also an inverse of itself? but I think it may not be!

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    You can achieve this by noting that not every element of $(Z,+_4)$ is the inverse of itself, and concluding that the two groups *cannot* be isomorphic.2011-12-05
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    @ Arturo;thank u,am okay now2011-12-05
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    It's bad for questions to remain unanswered. How about posting a proof that $V$ is not isomorphic to $\mathbb{Z}_4$ as an answer to your own question?2011-12-05
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    @neema You can prove that the only two groups of order 4 up to isomorphism are the Klein 4-group and the cyclic group of order 4. Since $(\mathbb{Z}_4, +_4)$ is isomorphic to the cyclic group of order 4 it cannot be isomorphic to the Klein 4-group.2011-12-05

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