Dihedral Group D4/Subgroups/Cosets/Subgroup Generated by b/Left Cosets
Jump to navigation
Jump to search
Examples of Left Cosets of Subgroups of Dihedral Group $D_4$
Let the dihedral group $D_4$ be represented by its group presentation:
- $D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$
Let $H \subseteq D_4$ be defined as:
- $H = \gen b$
where $\gen b$ denotes the subgroup generated by $b$.
From Subgroups of Dihedral Group D4 we have:
- $\gen b = \set {e, b}$
The left cosets of $H$ are:
\(\ds e H\) | \(=\) | \(\ds \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds b H\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds a H\) | \(=\) | \(\ds \set {a, b a^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds b a^3 H\) |
\(\ds a^2 H\) | \(=\) | \(\ds \set {a^2, b a^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds b a^2 H\) |
\(\ds a^3 H\) | \(=\) | \(\ds \set {a^3, b a}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds b a H\) |
Proof
The Cayley table of $D_4$ is presented as:
- $\begin{array}{l|cccccccc}
& e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\
\hline
e & e & a & a^2 & a^3 & b & b a & b a^2 & b a^3 \\ a & a & a^2 & a^3 & e & b a^3 & b & b a & b a^2 \\ a^2 & a^2 & a^3 & e & a & b a^2 & b a^3 & b & b a \\ a^3 & a^3 & e & a & a^2 & b a & b a^2 & b a^3 & b \\ b & b & b a & b a^2 & b a^3 & e & a & a^2 & a^3 \\ b a & b a & b a^2 & b a^3 & b & a^3 & e & a & a^2 \\ b a^2 & b a^2 & b a^3 & b & b a & a^2 & a^3 & e & a \\ b a^3 & b a^3 & b & b a & b a^2 & a & a^2 & a^3 & e
\end{array}$
Thus:
\(\ds e H\) | \(=\) | \(\ds e \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {e^2, e b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds b H\) | \(=\) | \(\ds b \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {b e, b^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {b, e}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds a H\) | \(=\) | \(\ds a \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a e, a b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a, b a^3}\) |
\(\ds b a^3 H\) | \(=\) | \(\ds b a^3 \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {b a^3 e, b a^3 b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {b a^3, a}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a H\) |
\(\ds a^2 H\) | \(=\) | \(\ds a^2 \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a^2 e, a^2 b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a^2, b a^2}\) |
\(\ds b a^2 H\) | \(=\) | \(\ds b a^2 \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {b a^2 e, b a^2 b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {b a^2, a^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^2 H\) |
\(\ds a^3 H\) | \(=\) | \(\ds a^3 \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a^3 e, a^3 b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a^3, b a}\) |
\(\ds b a H\) | \(=\) | \(\ds b a \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {b a e, b a b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {b a, a^3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^3 H\) |
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Exercise $4$