Normality Relation is not Transitive/Proof 2

Proof
Proof by Counterexample:

Let $D_4$ denote the dihedral group $D_4$.

Let $D_4$ be presented in matrix representation:

Its Cayley table is given as:

Consider the subgroup $H$ whose underlying set is:
 * $H = \set {\mathbf I, \mathbf A, -\mathbf I, -\mathbf A}$

From Subgroup of Index 2 is Normal, $H$ is normal in $D_4$.

Consider the subgroup $H$ whose underlying set is:
 * $K = \set {\mathbf I, \mathbf A} = \gen {\mathbf A}$

From Subgroup of Index 2 is Normal, $K$ is normal in $H$.

It remains to be demonstrated that $K$ is not normal in $D_4$.

From the Cayley table:
 * $\mathbf C \mathbf A \mathbf C^{-1} = \mathbf B \mathbf C = -\mathbf A \notin K$

Hence $K$ is not normal in $D_4$.