Subset Product of Subgroups/Examples/Subgroups Generated by b and a b in D3
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Examples of Use of Subset Product of Subgroups
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 $D_3$.
Proof
Consider the subset product $H K$:
\(\ds H K\) | \(=\) | \(\ds \set {h k: h \in H, k \in K}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {e, b, a b, b \paren {a b} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {e, b, a b, b \paren {b a^{-1} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {e, b, a b, a^{-1} }\) | as $b^2 = e$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {e, b, a b, a^2}\) |
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:
\(\ds K H\) | \(=\) | \(\ds \set {k h: k \in K, h \in H}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {e, a b, b, \paren {a b} b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {e, b, a b, a}\) | as $b^2 = e$ | |||||||||||
\(\ds \) | \(\ne\) | \(\ds H K\) |
So $H K \ne K H$ and so $H$ and $K$ are not permutable.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Exercise $5$