Klein four-group is no cyclic

**Rapha** · November 16th 2008, 06:58 AM

Proof:
The Klein four-group (tableau: ) $\begin{pmatrix}\circ & e & a & b & c\\ \hline e&e&a&b&c\\ a&a&e&c&b\\ b&b&c&e&a\\ c&c&b&a&e \end{pmatrix}$ 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

**clic-clac** · November 16th 2008, 07:09 AM

Look at the order of each element!

**Rapha** · November 16th 2008, 07:11 AM

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?

**clic-clac** · November 16th 2008, 07:32 AM

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

**Rapha** · November 16th 2008, 06:03 PM

Hi clic-clac,

Originally Posted by clic-clac

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

Merci beaucoup

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