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

## Examples of Cosets

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:

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