Rearranging the Cayley Table of the Dihedral Group by Cosets
A subgroup of a group is normal in if and only if every left coset of in is also a right coset of in . In such cases, cosets of in form a group, called the factor group . This Demonstration illustrates these results for the dihedral group by rearranging and coloring the elements in the Cayley table of the group by cosets.
The thumbnail and snapshot 1 illustrate the case where the cosets of a normal subgroup form a group. Snapshots 2 and 3 illustrate the case where the subgroup is not normal. In the latter case, coset multiplication is not well defined for either left or right cosets.
Notation and conventions follow Gallian's text . Rotations are taken as counterclockwise. Composition is from the right; in composing operations such as , the rotation is performed first.
 J. Gallian, Contemporary Abstract Algebra, 8th ed., Boston, MA: Brooks/Cole, 2013.