Symmetric Groups

Symmetric groups are groups based on symmetries of certain mathematical objects and structures.

Dihedral groups

An important example of a group comes from symmetries of geometric objects, most notably, polygons.

Lets take a hexagon, and apply an operation s, which reflects it along a line of symmetry. We could also apply an operation r, which rotates the hexagon by \pi / 3 radians. These operations swap the location of the vertices of the hexagon.

If we represent the orignal labelling of the vertices as

\{1,2,3,4,5,6\}

Then we can represent our operations r and s as the permutations

r = \{2,3,4,5,6,1\} \quad s = \{1,6,5,4,3,2\}

Definition

A symmetry is a reordering of the vertices on a geometric shape such that their structure remains. We typically denote a symmetry with \sigma.

For polygons in 2-space, we say that the ‘structure’ of the polygon remains if it can be transferred to 3-space, rotated, and then placed back into 2-space.

Thus our operations r and s are symmetries of the polygon. They can be combined to produce additional symmetries. The set of all possible symmetries on a n-polygon is denoted D_{2n}.

Definition

The set of symmetries on an n-polygon is denoted D_{2n}, and called the dihedral group of order 2n.

We will prove that D_{2n} is indeed a group later, but for now it can be taken as fact. D_{2n} has 2n symmetries, all of which can be constructed via compositions of our symmetries r and s.

Example

Lets examine the group D_{12}, the symmetries of a hexagon.

First lets look at our possible rotations r. Let e be the orignal labelling of the vertices, and r be defined

r \colon \{1,2,3,4,5,6\} \to \{2,3,4,5,6,1\}

We can repeatedly compose r to construct the following symmetries

e, r, r^2, r^3, r^4, r^5.

Note that r^6 = r^0 = e, thus |r| = 6.

Next we can look at our reflections. Let s be defined

s \colon \{1,2,3,4,5,6\} \to \{1,6,5,4,3,2\}

Note that s^2 = e. Combining s and r, we can create s, sr, sr^2, sr^3, sr^5, sr^6

Giving us six new symmetries. This gives us a total of 12 symmmetries in D_{12}.

It should be noted that D_{12} is not abelian, which can be clearly be seen with

rs = \{6,5,4,3,2,1\} = sr^5 \quad sr = \{2,1,6,5,4,3\}

Symmetric Groups

We may expand our notion of symmetry to include all permutations of the same collection of elements. This creates the more general symmetric group S_\Omega

Definition

Let \Omega is a non-empty set. S_\Omega is the set of all bijections from \Omega \to \Omega

These bijections are called permutations any typically notated \sigma or \tau

Note the following properties of S_\Omega

  • The identity of S_\Omega is the permutation e where

e(a) = a\quad \forall a \in \Omega

  • Every permutation \sigma \colon \Omega \to \Omega neccessarily has an inverse \sigma^{-1}, because \sigma is a bijection.

  • The composition of any two bijections \sigma \circ \tau is itself a bijection, and therefore in S_\Omega

Therefore (S_\Omega, \circ) is a group and its called the symmetric group on \Omega. Usually we take \Omega as the first n natural numbers, and denote the symmetric group S_n.

Example

Lets examine \Omega = \{1,2,3\}.

Let \sigma,\tau \in S_3 be

\sigma \colon \{1,2,3\} \to \{2,3,1\} \tau \colon \{1,2,3\} \to \{2,1,3\}

We can find \sigma \circ \tau \sigma \circ \tau \colon \{1,2,3\} \to \{3,2,1\}

Which is clearly in S_3

This notation is obtrusive and can be simplified using cycle decomposition.

Definition

A cycle is a string of integers representing a permutation in S_n, denoted (a_1\,a_2\,\ldots\,a_n).

A cycle represents the permutation

\{a_1,a_2,\ldots,a_n\} \to \{a_2,a_3,\ldots,a_1\}

Cycles can be concatenated to greatly reduce the amount of space needed to represent a permutation. For example the permutation

\{1,2,3,4,5,6\} \to \{2,3,1,5,4,6\}

Can be represented as three cycles,

(1\,2\,3)(4\,5)(6)

Often times cycles of length one are omitted.

Example

Given S_3, we know there exist 3! = 6 total permutations of the set.

The identity permutation (1)(2)(3)

Three examples where one number is left untouched (1)(2\,3), (2)(1\,3), (3)(1\,2)

And two examples where every number is permuted

(1\,2\,3), (1\,3\,2)

For any \sigma \in S_n, the cycle decomposition of \sigma^{-1} is obtained by writing the numbers in cycle of the cycle decomposition of \sigma is in reverse order. For example

\sigma \colon (1\,2\,3),\quad \sigma^{-1} \colon (3\,2\,1).

Note that S_n is non-abelian for all n\geq3. That being said, disjoint cycles will commute. The order of an element \sigma \in S_n is the least common multiple of the lengths in it’s cycle decomposition. For example, the order of

\sigma = (1\,2\,3)(4\,5)(6)

Would be \text{lcm}(3,2), thus |\sigma| = 6.