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:
$\quad \begin{array}{r|rrrrrr} \struct {\Z_6, +_6} & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \hline \eqclass 0 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\ \eqclass 1 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 \\ \eqclass 2 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 \\ \eqclass 3 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 \\ \eqclass 4 6 & \eqclass 4 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 \\ \eqclass 5 6 & \eqclass 5 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 \\ \end{array}$
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 $K$.