The Klein four-group is abelian but not cyclic

The Klein four-group is abelian but not cyclic
Tag	`DX6X1D21FE`
URI	`m2://DX6X1D21FE`
Kind	example
Writing	W0 stub
MSC	20A05
Instance of	Abelian group
Not an instance of	Cyclic group (note: no element generates the group.), Cyclic group (note: each element generates a subgroup with at most two elements.)
Uses	Klein four-group, Order of an element

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Example

The Klein four-group is $V = \{e, a, b, c\}$ , where $a^2 = b^2 = c^2 = e$ and $ab = ba = c$ . The group is abelian. Each element that is not $e$ has order $2$ . Thus each element generates a subgroup with at most $2$ elements. No element generates all $4$ elements, so $V$ is not cyclic.

Remarks

Thus the converse of the result that cyclic groups are abelian is false. Up to isomorphism, $V$ and $\mathbb{Z}/4\mathbb{Z}$ are the only groups of order $4$ .

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Binary operation definition
Group concept
Commutativity definition
Direct product of groups definition
Laws of exponents in a group statement
Abelian group definition
Order of an element definition
Klein four-group object
The Klein four-group is abelian but not cyclic example

Sources

Web https://en.wikipedia.org/wiki/Klein_four-group, lead section

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