📂Abstract Algebra

Klein Four-Group

Definition ¹

For $V = \left\{ e, a, b, c \right\}$ and a binary operation $\cdot$, $\left< V , \ \cdot \ \right>$ is called the klein 4-group.

Explanation

As can be seen, since the number of elements is only $4$ including the identity, it does not possess particularly rich properties. However, because it involves little computation and has its own operation, it serves as a fairly good example for grasping the concept of a group. Properties such as $x \cdot x = e$, meaning that every element is its own inverse, or that it cannot be expressed as $\left< x \right>$, can be checked with ease. Rather than being useful for any particular purpose, it is better to think of it as a group that helps one understand group theory simply through its very existence.

As mentioned earlier, $V$ is the group with the fewest elements among finite groups that are not cyclic groups. For reference, the group with the fewest elements among finite groups that are not Abelian groups is the dihedral group $D_{3} = S_{3}$.