Identifying subgroups and their cosets

Let G be a group and consider its multiplication table. By exchanging rows and columns you can put the elements of a subset at the beginning. By definition it is a subgroup if the products are also within this subset. We can see this clearly by either

• coloring the elements of the subset with the same color, or
• resizing temporarily the group to the subgroup.

For instance, the subset {0, 1} is not a subgroup from Z/6Z, but the subset {0, 3} is a subgroup. We can see this from

or from

Note that by resizing again to the original size, you get the original group back, as long as you did not change the structure essentially (e.g. by deleting an element).

1. Determine all subgroups and their cosets of the dihedral group D₄ below. Here r stands for the rotation and s for a reflection.
2. Can you predict the subgroups of D₅ and D₆? You can obtain those groups by entering "D5" or "D6" after pushing the "open" button.
3. How about subgroups of S₃ or A₄? (Enter "S3" or "A4")