Symmetric group

Cayley Graph of a symmetrical group of order 4 (S₄)

In mathematics, the symmetric group over a set X, denoted by S_X, is the group formed by the bijective applications of X in itself, under the operation of composition of functions.

When X = {1,...,n} is a finite set, the group S_X is called a permutation group of n elements, and is denoted by S_n. The order of this group is n!, and it is not abelian for n≥3.

Cayley's theorem states that every group G is isomorphic to a subgroup of its symmetric group S_G. In the particular case that G is finite of order n, then G is isomorphic to a subgroup of S_n.

Composition of permutations

There are several ways to represent a permutation. We can write a permutation σ in the form of a matrix, placing the domain elements 1, 2, 3... in the first row, and the corresponding images σ(1), σ(2), σ(3), in the second....

Given two permutations, their composition is done following the usual rules for composition of functions:

Yeah.

σ σ =(123456324651){displaystyle sigma ={begin{pmatrix}1 fake2 fake3 fake4 fake5 fake6\3 fake4 fake4 fake4 fake6}{pmatrix}}}}}}

and

Δ Δ =(123456412536){displaystyle tau ={begin{pmatrix}1 fake2 fake3 fake4 fake5 fake6\4 fake1 fake2 fake5 fake6\\end{pmatrix}}}}}}}}

its composition is: Δ Δ σ σ =(123456215634){displaystyle tau circ sigma ={begin{pmatrix}1 fake2 fake2}4 fake5 fake62 dream1 fake5 fake6 fake4\\end{pmatrix}}}}}}

The calculation of the composition can be followed in a visual way, remembering that when composing functions you operate from right to left:

Presentation of the group of permutations of n elements

Generators

Let's remember a translation is a permutation that exchanges two elements and fixes the remaining ones. All permutation is decomposed as a transposition product. Thus, the whole of the transpositions forms a system that generates Sn{displaystyle S_{n}}. But it is possible to further reduce this system by restricting us to the transpositions of the way Δ Δ i=(i,i+1){displaystyle tau _{i}=(i,i+1)}. Indeed, for i we can break down any transposition in the form:

(i,j)=(i,i+1)(i+1,i+2)...... (j− − 2,j− − 1)(j− − 1,j)(j− − 2,j− − 1)...... (i+1,i+2)(i,i+1){displaystyle (i,j)=(i,i+1)(i+1,i+2)dots (j-2,j-1)(j-1,j)(j-2,j-1)dots (i+1,i+2)(i,i+1)}

Elementary relations

These generators allow you to define a presentation of the symmetric group, together with the relations:

• Δ Δ i2=1{displaystyle {tau}}^{2}=1,},
• 1,}" xmlns="http://www.w3.org/1998/Math/MathML">Δ Δ iΔ Δ j=Δ Δ jΔ Δ iYeah.日本語j− − i日本語▪1{displaystyle tau _{i}tau _{j}=tau _{j}tau _{i}{i}qquad {mbox{si }}{j}{j}=tau,}}1,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/309dd8f6e360dd137b512d2cdedd20b02507284f" style="vertical-align: -1.005ex; width:27.913ex; height:3.009ex;"/>,
• (Δ Δ iΔ Δ i+1)3=1.{displaystyle {(tau _{i}tau _{i+1}}}})^{3}=1.,}.

Other generators

It is also possible to use as generator system:

• The transpositions of the form (1 (i), with i Primary.
• The set consists of only two generators: the trasposition σ=(1 2) and cycle c=(1 2... n).

Conjugation classes

Remember that every permutation can be described as a product of disjoint cycles, and this decomposition is unique except for the order of the factors. The conjugation classes of S_n correspond to the structure of said cycle decomposition: two permutations are conjugated in S_n if and only if they are obtained as a composition of the same number of disjoint cycles of the same lengths. For example, in S₅, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugates; but (1 2 3)(4 5) and (1 2)(4 5) don't.

The group S₃, formed by the 6 permutations of three elements, has three conjugation classes, listed with their element numbers:

• Identity (abc → abc) (1)
• Permutations that exchange two elements (abc → acb, abc → bac, abc → cba) (3)
• The cyclical permutations of the 3 elements (abc → bca, abc → cab) (2)

The group S₄, consisting of the 24 permutations of 4 elements, has 5 conjugation classes:

• Identity (1)
• Permutations that exchange two elements (6)
• Permutations that exchange cyclically three elements (8)
• Cyclical permutations of the four elements (6)
• Permutations that exchange two elements with each other, and also the remaining two (3)

In general, each conjugation class in S_n will correspond to an integer partition of n and can be represented graphically by a diagram Young's. So, for example, the five partitions of 4 would correspond to the five conjugation classes listed above:

1. 1 + 1 + 1 + 1
2. 2 + 1 + 1
3. 3 + 1
4. 4
5. 2 + 2

Group representations

If we associate each permutation with its permutation matrix, we obtain a representation that in general is not irreducible.

Irreducible representations