# Thread: Klein four-group is no cyclic

1. ## Klein four-group is no cyclic

Proof:
The Klein four-group (tableau: )$\displaystyle \begin{pmatrix}\circ & e & a & b & c\\ \hline e&e&a&b&c\\ a&a&e&c&b\\ b&b&c&e&a\\ c&c&b&a&e \end{pmatrix}$ is not a cyclic group

First I thought it is not cyclic because the Klein four-group is not commutative, but according to the tableau my guess is wrong.

Any help is much appreciated.

Rapha

2. Look at the order of each element!

3. Originally Posted by clic-clac
Look at the order of each element!
The order of each element is 2, but there are 4 elements

thats why the group is not cyclic?

4. Yep! Let $\displaystyle G$ be a cyclic group, $\displaystyle \exists x \in G / <x>=G$
Then $\displaystyle x$ has the same order as $\displaystyle G$.

5. Hi clic-clac,

Originally Posted by clic-clac
Yep! Let $\displaystyle G$ be a cyclic group, $\displaystyle \exists x \in G / <x>=G$
Then $\displaystyle x$ has the same order as $\displaystyle G$.
Merci beaucoup

### why is the klein 4 group not cyclic

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