Group Direct Product/Examples

Examples of Group Direct Products

Cyclic Group $C_2$ by Itself

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}$

Cyclic Group $C_2$ by $C_3$

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}$

Cyclic Group $C_3$ by Itself

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

Let us represent $C_3$ 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 $\struct{C_3 \times C_3, +_9}$ can be portrayed as:

$\begin {array} {r|rrrrrrrrr}

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

Product of $\R \setminus \set 0$ by $\R$

Let $G$ be the Cartesian product of $\R \setminus \set 0$ with $\R$:

$G = \set {\tuple {a, b} \in \R^2: a \ne 0}$

Let $\circ$ be a group product on $G$ defined as:

$\tuple {a_1, b_1} \circ \tuple {a_2, b_2} = \tuple {a_1 a_2, a_1 b_2 + b_1}$

Then the algebraic structure $\struct {G, \circ}$ is a group which is non-abelian.