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 
, and an identity 
. 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:
![\[i^2=j^2=k^2=ijk=e.\]](http://latex.artofproblemsolving.com/5/4/b/54b8cf2a2473b9793cf37bbabb57bbd073457263.png)
The Klein 4-group is isomorphic to 
. It is also the group of symmetries of a rectangle. It has three isomorphic subgroups, each of which is isomorphic to 
, and, of course, is normal, since the Klein 4-group is abelian.
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