Symmetric Groups, Permutation Groups, and Dihedral Groups Review
We will now review some of the recent material regarding symmetric groups and dihedral
groups.
Recall from the The Symmetric Groups on n Elements page that for the n-element
set {1,2,...,n} that the set of all permutations σ on {1,2,...,n} is denoted Sn where
σ:{1,2,...,n}→{1,2,...,n} is a permutation on this set if σ is a bijection.
We then defined the Symmetric Group on n Elements to be (Sn,∘) where ∘ is the
operation of function composition defined for all f1,f2∈Sn by:
(1)
(f1∘f2)(x)=f1(f2(x))
We also saw that the order of (Sn,∘) is n!, i.e., there exists n! permutations σ on the set
{1,2,...,n}.
On the The Symmetric Group of a General n-Element Set page we extended the
notion of a symmetric group to any n-element set A={x1,x2,...,xn} analogously above.
We denote the set of all permutations on A by SA and the Symmetric Group on A is
defined to be (SA,∘).
We then proved a very simple theorem which says that if A is any finite set and
Ga⊆SA is the subset of permutations on A for which σ(a)=a then (Ga,∘) is a subgroup
of (SA,∘). It was fairly easy to check that Ga is closed under ∘ and that for every σ∈Ga
we have that σ−1∈Ga.
On the Permutation Groups on a Set we said that if X is ANY set and SX denotes the
set of all permutations on X then the Permutation Group on X is (SX,∘). If
X={1,2,...,n} then SX=Sn, i.e., the symmetric groups on n-elements are permutation
groups!
On The Dihedral Groups Dn page we began to look at groups known as Dihedral
Groups (Dn,∘) defined for all integers n≥3 where Dn is the set of all permutations
which are symmetries of the regular n-gon (the regular n-sided polygon) and ∘ is the
operation of function composition.
We then examined some of these dihedral groups on the following pages:
o The Group of Symmetries of the Equilateral Triangle.
o The Group of Symmetries of the Square.
o The Group of Symmetries of the Pentagon.
On The Group of Symmetries of a Rectangle page we then looked at the group of
symmetries of a nonregular polygon - the rectangle. Identifying the symmetries for
this group was analogous to that of the regular polygons above.
The dihedral group is the symmetry group of an -sided regular polygon for .
The group order of is . Dihedral groups are non-Abelian permutation groups
for .
The th dihedral group is represented in the Wolfram Language as
DihedralGroup[n].
One group presentation for the dihedral group is .
A reducible two-dimensional representation of using real matrices has generators
given by and , where is a rotation by radians about an axis passing through the
center of a regular -gon and one of its vertices and is a rotation by about the
center of the -gon (Arfken 1985, p. 250).
Dihedral groups all have the same multiplication table structure. The table for is
illustrated above.
The cycle index (in variables , ..., ) for the dihedral group is given by
(1)
where
(2)
is the cycle index for the cyclic group , means divides , and is the totient
function (Harary 1994, p. 184). The cycle indices for the first few are
(3)
(4)
(5)
(6)
(7)
Renteln and Dundes (2005) give the following (bad) mathematical joke about the
dihedral group:
Q: What's hot, chunky, and acts on a polygon? A: Dihedral soup
GRUP SIMETRI
Sekarang kita akan melihat permutasi-permutasi dari himpunan berhingga
, .
Misal diambil contoh untuk . Perhatikan semua kemungkinan permutasi dari sebagai
berikut:
