# Group Direct Product/Examples/C2 x C2

## Example of Group Direct Product

The direct product of $C_2$, the cyclic group of order $2$, with itself is as follows.

Let us represent $C_2$ as the group $\struct {\set {1, -1}, \times}$:

$\begin {array} {r|rr} \struct {\set {1, -1} , \times} & 1 & -1 \\ \hline 1 & 1 & -1 \\ -1 & -1 & 1 \\ \end{array}$

Then the Cayley table for $C_2 \times C_2$ can be portrayed as:

$\begin {array} {c|cccc} C_2 \times C_2 & \tuple { 1, 1} & \tuple { 1, -1} & \tuple {-1, 1} & \tuple {-1, -1} \\ \hline \tuple { 1, 1} & \tuple { 1, 1} & \tuple { 1, -1} & \tuple {-1, 1} & \tuple {-1, -1} \\ \tuple { 1, -1} & \tuple { 1, -1} & \tuple { 1, 1} & \tuple {-1, -1} & \tuple {-1, 1} \\ \tuple {-1, 1} & \tuple {-1, 1} & \tuple {-1, -1} & \tuple { 1, 1} & \tuple { 1, -1} \\ \tuple {-1, -1} & \tuple {-1, -1} & \tuple {-1, 1} & \tuple { 1, -1} & \tuple { 1, 1} \\ \end{array}$

This is seen by inspection to be an instance of the Klein $4$-group.

## Subgroups

The subsets of $C_2 \times C_2$ which form its subgroups are:

$\set {\tuple {1, 1} }$
$\set {\tuple {1, 1}, \tuple {1, -1} }$
$\set {\tuple {1, 1}, \tuple {-1, 1} }$
$\set {\tuple {1, 1}, \tuple {-1, -1} }$
$\set {\tuple {1, 1}, \tuple {1, -1}, \tuple {-1, 1} , \tuple {-1, -1} }$ (that is, $C_2 \times C_2$ itself)