Rearranging the Cayley Table of the Dihedral Group by Cosets

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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.
Contributed by: Marc Brodie (March 2011)
(Benedictine University Mesa)
Open content licensed under CC BY-NC-SA
Details
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 [1]. Rotations are taken as counterclockwise. Composition is from the right; in composing operations such as , the rotation is performed first.
Reference
[1] J. Gallian, Contemporary Abstract Algebra, 8th ed., Boston, MA: Brooks/Cole, 2013.
Snapshots
Permanent Citation