Group Direct Product/Examples/C2 x C3

Example of Group Direct Product

The direct product of $C_2$, the cyclic group of order $2$, with $C_3$, the cyclic group of order $3$, is as follows.

Let us represent $C_2$ as the group $\struct {\Z_2, +_2}$:

$\begin {array} {r|rr}

+_2 & \eqclass 0 2 & \eqclass 1 2 \\ \hline \eqclass 0 2 & \eqclass 0 2 & \eqclass 1 2 \\ \eqclass 1 2 & \eqclass 1 2 & \eqclass 0 2 \\ \end{array}$

and $C_3$ as the group $\struct {\Z_3, +_3}$:

$\begin {array} {r|rrr}

+_3 & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \hline \eqclass 0 3 & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \eqclass 1 3 & \eqclass 1 3 & \eqclass 2 3 & \eqclass 0 3 \\ \eqclass 2 3 & \eqclass 2 3 & \eqclass 0 3 & \eqclass 1 3 \\ \end{array}$

Then the Cayley table for $\struct{C_2 \times C_3, +_6}$ can be portrayed as:

$\begin {array} {r|rrrrrr}

+_6 & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} \\ \hline \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} \\ \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} \\ \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} \\ \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} \\ \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 0 3} \\ \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 2 3} & \tuple {\eqclass 1 2, \eqclass 0 3} & \tuple {\eqclass 1 2, \eqclass 1 3} & \tuple {\eqclass 0 2, \eqclass 2 3} & \tuple {\eqclass 0 2, \eqclass 0 3} & \tuple {\eqclass 0 2, \eqclass 1 3} \\ \end{array}$

Sources

• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $1$: Definitions and Examples: Exercise $5$