Subset Product of Subgroups/Examples/Subgroups Generated by b and a b in D3

Examples of Use of Subset Product of Subgroups
Consider the dihedral group $D_3$, given as the group presentation:

Consider the generated subgroups $H := \gen b$ and $K := \gen {a b}$:

Then $H$ and $K$ are not permutable, and neither $H K$ nor $K H$ is a subgroup of $D_3$.

Proof
Consider the subset product $H K$:

But $\set {e, b, a b, a^2}$ has $4$ elements.

Thus by Lagrange's Theorem (Group Theory), $H K$ is not a subgroup of $D_3$.

Then we see:

So $H K \ne K H$ and so $H$ and $K$ are not permutable.