Dihedral Group D4 is not Internal Group Product

Theorem

The dihedral group $D_4$ is not an internal group product of any $2$ of its proper subgroups.

Proof 1

Aiming for a contradiction, suppose $D_4$ is the internal group product of $2$ proper subgroups $H$ and $K$ of $D_4$.

Without loss of generality, let $\order H = 2$ and $\order K = 4$.

$H$ needs to be normal in $D_4$ for the conditions of the internal group product to be satisfied.

So the non-identity element of $H$ needs to be conjugated to itself by every element of $G$.

This means $H$ is a subset of the center of $G$.

From Center of Dihedral Group $D_4$:

$H = \set {e, a^2}$

But every non-trivial normal subgroup of $D_4$ contains $a^2$.

This makes it impossible for $H \cap K = \set e$.

Thus $D_4$ cannot be an internal group product.

$\blacksquare$

Proof 2

The proper subgroups of $D_4$ are of order no greater than $4$.

From Group of Order less than 6 is Abelian, all such proper subgroups are abelian.

From External Direct Product of Abelian Groups is Abelian Group, the group direct product of $2$ of these proper subgroups is in turn abelian.

From Internal and External Group Direct Products are Isomorphic, the internal group product of $2$ of these proper subgroups is in turn abelian.

But $D_4$ is not abelian.

Hence it cannot be the internal group product of any $2$ of these proper subgroups.

$\blacksquare$