Klein 4-group

The Klein 4-group is an Abelian group. It is the smallest non-cyclic group. It is the underlying group of the four-element field.

The Klein 4-group consists of three elements $i,j,k$, and an identity $e$. Every element is its own inverse, and the product of any two distinct non-identity elements is the remaining non-identity element. Thus the Klein 4-group admits the following elegant presentation: \[i^2=j^2=k^2=ijk=e.\] The Klein 4-group is isomorphic to $(\mathbb{Z}/2\mathbb{Z})\times (\mathbb{Z}/2\mathbb{Z})$. It is also the group of symmetries of a rectangle. It has three isomorphic subgroups, each of which is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, and, of course, is normal, since the Klein 4-group is abelian.

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