MATH 57091 - Algebra for High School Teachers
Dihedral & Symmetric Groups
Professor Donald L. White
Department of Mathematical Sciences
Kent State University
D.L. White (Kent State University)
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Dihedral Groups
The group of symmetries of the square is the dihedral group of order 8,
denoted D4 . Recall that D4 contains 4 rotations and 4 reflections.
More generally, a regular n-gon has
n rotations through multiples of
360
n
degrees, and
n reflections across its n axes of symmetry.
Definition
The group of symmetries of the regular n-gon (for n > 3)
is called the dihedral group of order 2n and is denoted Dn .
NOTE: Groups are sometimes named for their order (number of elements)
and sometimes for the size of the object they “act” on.
Accordingly, some sources use Dn to denote the dihedral group of order 2n,
(as in our text) while others use D2n . Thus, depending on the source,
the group of symmetries of the square may be called D4 or D8 .
D.L. White (Kent State University)
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Symmetries of the Pentagon
Example 1: Symmetries of the regular pentagon.
The central angle of a regular pentagon is
360
5
degrees; that is, 72◦ :
Therefore, the dihedral group D5 contains 5 rotations,
through 0◦ , 72◦ , 144◦ , 216◦ , and 288◦ . For example, the 144◦ rotation
-
corresponds to the permutation of vertices
1 2 3 4 5
.
3 4 5 1 2
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Symmetries of the Pentagon
[Example 1, continued]
The pentagon has no pairs of opposite sides or pairs of opposite vertices.
In this case, each axis of symmetry is a line through a vertex
and the center of the opposite side:
For example, the reflection across the axis through vertex 2 is
-
D.L. White (Kent State University)
or
1 2 3 4 5
3 2 1 5 4
.
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Symmetries of the Hexagon
Example 2: Symmetries of the regular hexagon.
The central angle of a regular hexagon is
360
6
degrees; that is, 60◦ :
Therefore, the dihedral group D6 contains 6 rotations,
through 0◦ , 60◦ , 120◦ , 180◦ , 240◦ , and 300◦ ; e.g., the 120◦ rotation
-
corresponds to the permutation of vertices
1 2 3 4 5 6
.
3 4 5 6 1 2
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Symmetries of the Hexagon
[Example 2, continued]
The hexagon has 3 pairs of opposite vertices and 3 pairs of opposite sides.
Each axis of symmetry is a line through one of these pairs.
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Symmetries of the Hexagon
[Example 2, continued]
For example, the reflection across the axis through vertices 2 and 5 is
-
or
1 2 3 4 5 6
3 2 1 6 5 4
.
The reflection across the horizontal axis of symmetry is
-
D.L. White (Kent State University)
or
1 2 3 4 5 6
4 3 2 1 6 5
.
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Symmetric Groups
For n > 4, not every permutation of the vertices of an n-gon is in Dn .
For example, if n = 5, then the permutation
1 2 3 4 5
1 5 3 4 2
of the vertices of
is not a rigid motion, hence is not in D5 .
Therefore, for n > 4,
Dn is a proper subset of the set of all permutations of {1, 2, 3, . . . , n}
(i.e., the set of one-to-one, onto functions from {1, 2, 3, . . . , n} to itself).
Definition
The set of all permutations of the set {1, 2, 3, . . . , n} is the
symmetric group on n letters, denoted Sn .
D.L. White (Kent State University)
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Symmetric Groups
As before, we can write the permutations in Sn in “two-row” form:
1 2 3 4 ··· n
,
a1 a2 a3 a4 · · · an
where a1 , a2 , . . . , an are the integers 1, 2, . . . , n in some order.
Since all one-to-one, onto functions are permitted, there are
n
n−1
n−2
..
.
choices for
choices for
choices for
..
.
a1
a2
a3
..
.
2
1
choices for
choice for
an−1
an .
All choices are independent, so the total number of choices is
n(n − 1)(n − 2) · · · (2)(1) = n!,
hence |Sn | = n!. As |Dn | = 2n, we have Dn ( Sn for n > 4.
Note, however, that 3! = 2 · 3, so in fact D3 = S3 .
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Symmetric Groups
We can again combine two permutations by function composition.
For example, if
1 2 3 4 5 6 7
1 2 3 4 5 6 7
f =
and g =
2 3 5 4 6 7 1
6 5 2 1 4 7 3
in S7 , then (composing right to left as usual)
1 2 3 4 5 6 7
1 2 3 4 5 6 7
f ◦g =
◦
2 3 5 4 6 7 1
6 5 2 1 4 7 3
=
1 2 3 4 5 6 7
7 6 3 2 4 1 5
.
Composition of functions is always associative.
Therefore, this binary operation on Sn is associative.
D.L. White (Kent State University)
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Symmetric Groups
The permutation
i=
1 2 3 4 ···
1 2 3 4 ···
n
n
is an identity element for Sn ; that is, f ◦ i = i ◦ f = f for all f in Sn .
Every one-to-one, onto function f has an inverse function f −1 ;
that is, f ◦ f −1 = f −1 ◦ f = i.
Therefore, every element of Sn has an inverse in Sn .
The inverse of f ∈ Sn is easy to find: turn the two-row form upside down.
For example, in S7 , if
1 2 3 4 5 6 7
f =
,
7 6 3 2 4 1 5
then
f −1 =
7 6 3 2 4 1 5
1 2 3 4 5 6 7
D.L. White (Kent State University)
=
1 2 3 4 5 6 7
6 4 3 5 7 2 1
.
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