Internal Direct Product Theorem/Examples

Examples of Use of Internal Direct Product Theorem

Symmetry Group of Rectangle

Consider the symmetry group of the rectangle $D_2$:

Let $\RR = ABCD$ be a (non-square) rectangle.

The various symmetry mappings of $\RR$ are:

The identity mapping $e$

The rotation $r$ (in either direction) of $180^\circ$

The reflections $h$ and $v$ in the indicated axes.

The symmetries of $\RR$ form the dihedral group $D_2$.

Let $H := \set {e, r}$.

Let $K := \set {e, h}$.

Then $H$ and $K$ are subgroups of $D_2$ which fulfil the conditions of the Internal Direct Product Theorem, as:

$r \circ h = v = h \circ r$

Thus $D_2$ is the internal group direct product of $H$ and $K$.

Both $H$ and $K$ are isomorphic to $\struct {\Z_2, +_2}$, the additive group of integers modulo $2$.

Hence by Isomorphism of External Direct Products:

$D_2$ is isomorphic to $\struct {\Z_2, +_2} \times \struct {\Z_2, +_2}$.

Additive Group of Integers Modulo $6$

Consider the additive group of integers modulo $6$ $\struct {\Z_6, \times_6}$, illustrated by Cayley Table:

$\begin{array}{r|rrrrrr}

\struct {\Z_6, +_6} & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \hline \eqclass 0 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \eqclass 1 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 \\ \eqclass 2 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 \\ \eqclass 3 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 \\ \eqclass 4 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 \\ \eqclass 5 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 \\ \end{array}$

Let $H := \set {0, 2, 4}$.

Let $K := \set {0, 3}$.

We have that:

$H +_6 K = \struct {\Z_6, +_6}$

and:

$H \cap K = \set 0$

Hence $H$ and $K$ are subgroups of $\struct {\Z_6, +_6}$ which fulfil the conditions of the Internal Direct Product Theorem.

Thus $\struct {\Z_6, +_6}$ is the internal group direct product of $H$ and $K$.

Because:

$H$ is isomorphic to $\struct {\Z_3, +_3}$

$K$ is isomorphic to $\struct {\Z_2, +_2}$

it follows by Isomorphism of External Direct Products that:

$\struct {\Z_6, +_6}$ is isomorphic to $\struct {\Z_3, +_3} \times \struct {\Z_2, +_2}$.

Multiplicative Monoid of Integers Modulo $6$

Consider the multiplicative monoid of integers modulo $6$ $\struct {\Z_6, +_6}$, illustrated by Cayley Table:

$\quad \begin{array} {r|rrrrrr} \struct {\Z_6, \times_6} & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \hline \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 \\ \eqclass 1 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \eqclass 2 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 \\ \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 \\ \eqclass 4 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 \\ \eqclass 5 6 & \eqclass 0 6 & \eqclass 5 6 & \eqclass 4 6 & \eqclass 3 6 & \eqclass 2 6 & \eqclass 1 6 \end{array}$

Let $H := \set {0, 2, 4}$.

Let $K := \set {0, 3}$.

We have that:

$H \times_6 K = \set 0$

so $\struct {\Z_6, \times_6}$ is not the internal group direct product of $H$ and $K$.