Cosets of a group

This blog is about just putting one simple example of calculating left/right cosets of a group. In case you are interested in theory and theorems as well than please visit other sites though for convenience I will put one definition from wiki here.

If G is a group, and H is a subgroup of G, and g is an element of G, then

gH = {gh : h an element of H } is a left coset of H in G, and

Hg = {hg : h an element of H } is a right coset of H in G.

Example:

Let G be a symmetric group of order 3 so elements of group are as,

Group Elements i.e 'g' = {{ },{1,2},{2,3},{1,3},{1,2,3},{3,2,1}} 

There will be six subgroups to this group and to calculate cosets lets pick one such subgroup.So,

                          H= {{ },{1,2}}

Left Cosets: Now to calculate its cosets just apply permutation composition/multiplication with rest of the elements of group on the left as follows.

1. { }{ }{1,2} = {{ },{1,2}}.

2. {1,2}{ }{1,2} = {{1,2},{ }}.

3. {1,3}{ }{1,2} = {{1,3},{1,2,3}}.

4. {2,3}{ }{1,2} = {{2,3},{1,3,2}}.

5. {1,2,3}{ }{1,2} = {{1,2,3},{1,3}}.

6. {1,3,2}{ }{1,2} = {{1,3,2},{2,3}}.

Right Cosets: For calculating right cosets multiply/compose the subgroup as a whole with elements of group on right hand side.

1.{ }{1,2}{ } = {{ },{1,2}}.

2.{ }{1,2}{1,2} = {{1,2},{ }}.

3.{ }{1,2}{1,3} = {{1,3},{1,3,2}}.

4.{ }{1,2}{2,3} = {{2,3},{1,2,3}}.

5.{ }{1,2}{1,3,2} = {{1,3,2},{1,3}}.

6.{ }{1,2}{1,2,3} = {{1,2,3},{2,3}}.

Hope this tutorial helps you in understanding logic behind calculating cosets.