Internal Group Direct Product/Examples/D4

Example of Internal Group Direct Product
Consider the dihedral group $D_4$, which is the symmetry group of the square.

Suppose $D_4$ is the internal group direct product of two subgroups.

Then those two subgroups are $\set e$ and $D_4$ itself, where $e$ is the identity element of $D_4$.

Proof
Let $H$ and $K$ be subgroups of $D_4$ such that $D_4$ is the internal group direct product of $H$ and $K$.

Then from the Internal Direct Product Theorem $H$ and $K$ must be:
 * normal subgroups
 * whose intersection is $\set e$
 * whose subset product is $D_4$.

Let $D_4$ be expressed in its group presentation:

Pairs of subgroups of $D_4$ whose intersection is $\set e$ and whose subset product is $D_4$ are:


 * $(1): \quad \set e$ and $D_4$


 * $(2): \quad \set {e, b}$ and $\set {e, a, a^4, a^3}$


 * $(3): \quad \set {e, b a}$ and $\set {e, a^2, b, b a^2}$


 * $(3): \quad \set {e, b a^2}$ and $\set {e, a^2, b, b a^3}$


 * $(5): \quad \set {e, b a^3}$ and $\set {e, a^2, b, b a^2}$

From Normal Subgroups of Dihedral Group, none of $\set {e, b}$, $\set {e, b a}$, $\set {e, b a^2}$ or $\set {e, b a^3}$ are normal.

Hence the result.