Coset/Examples/Symmetry Group of Equilateral Triangle/Cosets of Reflection Subgroup
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Examples of Cosets
Consider the symmetry group of the equilateral triangle $D_3$.
Let $H \subseteq D_3$ be defined as:
- $H = \set {e, r}$
where:
- $e$ denotes the identity mapping
- $r$ denotes reflection in the line $r$.
The left cosets of $H$ are:
| \(\ds H\) | \(=\) | \(\ds \set {e, r}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds e H\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds r H\) | ||||||||||||
| \(\ds s H\) | \(=\) | \(\ds \set {s e, s r}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \set {s, q}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds q H\) | ||||||||||||
| \(\ds t H\) | \(=\) | \(\ds \set {t e, t r}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \set {t, p}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds p H\) |
The right cosets of $H$ are:
| \(\ds H\) | \(=\) | \(\ds \set {e, r}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds H e\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds H r\) | ||||||||||||
| \(\ds H s\) | \(=\) | \(\ds \set {e s, r s}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \set {s, p}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds H p\) | ||||||||||||
| \(\ds H t\) | \(=\) | \(\ds \set {e t, r t}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \set {t, q}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds H q\) |
Transversals
Some of the left transversals of $H$ are given by:
- $\set {e, s, t}$
- $\set {e, q, p}$
- $\set {r, s, p}$
and so on.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.1$. The quotient sets of a subgroup: Example $112$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 42$. Another approach to cosets: Worked Example $1$