Classifications of Small Groups

group-theorynote

Groups with one element

Since all groups must contain $e$, this is the trivial group,

$$G=\{g\}\to G=\{e\}.$$

Groups with two elements

Must contain $e$ and one more element $g$. Therefore, $g=g^{-1}$,

$$G=\{e,g\}e\ne g,g=g^{-1}.$$

Example

$({\mathbb{Z}}_2,+)$ is a group with two elements.

Related Theorem

Theorem - The inverse of the identity element is itself

Groups with three elements

$$G=\{e,g,g^{-1}\}.$$

Groups with four elements

There are two types of groups with 4 elements.

Isomorphic to $({\mathbb{Z}}_4,+)$

$$\begin{array}{ccccc}+& 0& 1& 2& 3\\ 0& 0& 1& 2& 3\\ 1& 1& 2& 3& 0\\ 2& 2& 3& 0& 1\\ 3& 3& 0& 1& 2\end{array}$$

Isomorphic to $({\mathbb{Z}}_2\times {\mathbb{Z}}_2,+)$

• Also called the Klein four-group.
• Every element is its own inverse.

$$\begin{array}{ccccc}+& (0,0)& (0,1)& (1,0)& (1,1)\\ (0,0)& (0,0)& (0,1)& (1,0)& (1,1)\\ (0,1)& (0,1)& (0,0)& (1,1)& (1,0)\\ (1,0)& (1,0)& (1,1)& (0,0)& (0,1)\\ (1,1)& (1,1)& (1,0)& (0,1)& (0,0)\end{array}$$

Commutativity

Theorem - The smallest non-abelian group has order 6.