Dihedral Group of the Square

​
labels
1
2
3
subgroup
1
There are 10 subgroups. This subgroup is not abelian.
set =
{
I
4
,
R
1
,
R
2
,
R
3
,H,
Δ
1
,V,
Δ
2
}
inverse =
{
I
4
,
R
3
,
R
2
,
R
1
,H,
Δ
1
,V,
Δ
2
}
order =
{1,4,2,4,2,2,2,2}
In mathematics, a dihedral group
D
n
is the group of symmetries of a regular polygon with
n
sides, including both rotations and reflections. This Demonstration shows the subgroups of
D
4
, the dihedral group of a square.

Details

A group is a set
G
together with a binary operation
•
on
G
, i.e., a function
•:G×G
to
G
(called the group law of
G
) that combines any two elements
a
and
b
to form another element, denoted
a•b
or
ab
. To qualify as a group, the set and operation, (
G
,
•
), must satisfy four requirements known as the group axioms: closure, associativity, identity element, and inverse element. If
a•b=b•a
, then the group is commutative or Abelian.
In this Demonstration, the group law is the composition of permutations of the set
{1,2,3,4}
. For example,

1
2
3
4
2
3
4
1
•
1
2
3
4
1
4
3
2
=
1
2
3
4
2
1
4
3

.

Permanent Citation

Gerard Balmens
​
​"Dihedral Group of the Square"​
​http://demonstrations.wolfram.com/DihedralGroupOfTheSquare/​
​Wolfram Demonstrations Project​
​Published: January 27, 2014