# Cyclic group of order $4$ and Klein four-group are not isomorphic. [duplicate]

Possible Duplicate:
The Klein 4-group vs. the integers modulo 4

Prove that the cyclic group of order $4$ and the Klein four-group are not isomorphic.

Can someone explain what Klein four-group is and how to do this question?

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## marked as duplicate by YACP, Did, Ittay Weiss, Stefan Hansen, sdcvvcJan 20 '13 at 9:59

In what context did you encounter the problem? Do you have a book or notes? Was the definition not given? When you want to look something up quickly and don't have a reference book/notes handy, it often helps to Google key words, such as Klein four group, which will lead you quickly to a reference such as Wikipedia. – Jonas Meyer Jan 20 '13 at 8:36
Klein four group is $\mathbb{Z}_2 \times \mathbb{Z}_2$, and every elements satisfy the equation $2x=0$, but for $\mathbb{Z}_4$, it's not true. ($1+1 \neq 0$) so they can't be isomorphic.
The Klein four group is $$\Bbb Z_2\times\Bbb Z_2,$$ where $\Bbb Z_2$ is the cyclic group of $2$ elements. In this group, every element has order at most $2$, while in a cyclic group of order $4$, two elements have order $4$, hence they cannot be isomorphic.