Dihedral Group D4/Subgroups/Cosets/Subgroup Generated by b/Left Cosets

From ProofWiki
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

Retrieved from "https://proofwiki.org/w/index.php?title=Dihedral_Group_D4/Subgroups/Cosets/Subgroup_Generated_by_b/Left_Cosets&oldid=387876"