Subset Product of Subgroups/Examples

Examples of Use of Subset Product of Subgroups

Subgroups $\gen b$ and $\gen {a b}$ in $D_3$

Consider the dihedral group $D_3$, given as the group presentation:

$D_3 = \gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$

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

\(\ds \gen b\)

\(=\)

\(\ds \set {e, b}\)

as $b^2 = e$

\(\ds \gen {a b}\)

\(=\)

\(\ds \set {e, a b}\)

as $\paren {a b}^2 = a b b a^{-1} = e$

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