Normal subgroups and quotients

Visualising cosets in a group can be done by coloring the table. Put the elements of the subgroup to the left, and group for all cosets the elements together. If you get in the multiplication table a nice block structure, then the subgroup is a normal subgroup. We show this for S₃.

The structure of the quotient group can be read of from the picture. It is a group of order 2.

1. Determine all normal subgroups of D₄ and identify the quotient groups.
2. Determine all normal subgroups of D₆ and identify the quotient groups.
3. Determine all normal subgroups of A₄ and identify the quotient groups.