Klein Four-group
Definition 1
Given $V = \left\{ e, a, b, c \right\}$ and the binary operation $\cdot$, $\left< V , \ \cdot \ \right>$ is referred to as the Klein four-group.
Description
As you can see, since the number of elements is only $4$, including the identity element, it does not possess very rich properties. However, it serves as a very good example for getting acquainted with the concept of groups since it involves little calculation and has its unique operations. Properties such as every element being its own inverse like in $x \cdot x = e$ or not being represented like in $\left< x \right>$ can be easily verified without much trouble. Rather than being used for something in particular, it is better to think of it as a group that helps in understanding group theory simply by its existence.
As mentioned earlier, $V$ is the smallest group among finite groups that is not a cyclic group. For reference, the smallest finite group that is not an abelian group is the dihedral group of $D_{3} = S_{3}$.
Fraleigh. (2003). A first course in abstract algebra(7th Edition): p51. ↩︎