Coset/Examples/Symmetric Group on 3 Letters/Cosets of Alternating Subgroup

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Examples of Cosets


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Consider the symmetric group on 3 Letters.

Let $S_3$ denote the set of permutations on $3$ letters.


The symmetric group on $3$ letters is the algebraic structure:

$\struct {S_3, \circ}$

where $\circ$ denotes composition of mappings.


Let $H \subseteq S_3$ be defined as:

$H = \set {e, \tuple {1 2 3}, \tuple {1 3 2} }$

The cosets of $H$ are:

\(\ds e H\) \(=\) \(\ds \set {e, \tuple {1 2 3}, \tuple {1 3 2} }\)
\(\ds \) \(=\) \(\ds \tuple {1 2 3} H\)
\(\ds \) \(=\) \(\ds \tuple {1 3 2} H\)
\(\ds \) \(=\) \(\ds H\)
\(\ds \tuple {1 2} H\) \(=\) \(\ds \set {\tuple {1 2}, \tuple {1 2} \tuple {1 2 3}, \tuple {1 2} \tuple {1 3 2} }\)
\(\ds \) \(=\) \(\ds \set {\tuple {1 2}, \tuple {2 3}, \tuple {1 3} }\)


Sources

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