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.