Rearranging the Cayley Table of the Dihedral Group by Cosets

​
subgroup
〈
R
90
〉
〈V,H〉
〈D,D'〉
〈H〉
〈V〉
〈
R
180
〉
〈D〉
〈D'〉
cosets
left
right
rearrange elements in Cayley table by cosets
color elements in Cayley table by cosets
home screen
subgroup: {
R
0
,
R
180
,H,V}
left cosets:
{
R
0
,
R
180
,H,V}
{
R
90
,
R
270
,D,
′
D
}
right cosets:
{
R
0
,
R
180
,H,V}
{
R
90
,
R
270
,D,
′
D
}
A subgroup
H
of a group
G
is normal in
G
if and only if every left coset of
H
in
G
is also a right coset of
H
in
G
. In such cases, cosets of
H
in
G
form a group, called the factor group
G/H
. This Demonstration illustrates these results for the dihedral group
D
4
by rearranging and coloring the elements in the Cayley table of the group
D
4
by cosets.

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
H∘
R
90
, the rotation is performed first.

References

[1] J. Gallian, Contemporary Abstract Algebra, 8th ed., Boston, MA: Brooks/Cole, 2013.

Permanent Citation

Marc Brodie
​
​"Rearranging the Cayley Table of the Dihedral Group by Cosets"​
​http://demonstrations.wolfram.com/RearrangingTheCayleyTableOfTheDihedralGroupByCosets/​
​Wolfram Demonstrations Project​
​Published: December 28, 2018