What Are Symmetric Groups and Their Mathematical Significance?

In summary, the symmetric group S(n) or Sym(n) is the group of all possible permutations of n symbols. It has an index-2 subgroup, the alternating group A(n) or Alt(n), which is the group of all possible even permutations of n symbols. The alternating group is simple for n >= 5. The conjugacy classes of S(n) are determined by the lengths of cycles in the permutations, and the representation theory of the symmetric group is based on these lengths. Going from a symmetric group to its alternating group, the lengths of the cycles are either merged or split, depending on their duality. The character values for the alternating group are determined by this duality, with odd permutations getting a reversed sign. The
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Definition/Summary

The symmetric group S(n) or Sym(n) is the group of all possible permutations of n symbols. It has order n!.

It has an index-2 subgroup, the alternating group A(n) or Alt(n), the group of all possible even permutations of n symbols. That group has order n!/2. For n >= 5, the alternating group is simple.

Equations



Extended explanation

First, some details about permutations. Every finite-length permutation can be broken up into cyclic permutations, where a cyclic permutation has the form
a1 -> a2, a2 -> a3, a3 -> a4, ..., a(n) -> a1

where n is the length of the cycle.

The conjugacy classes of S(n) are all elements that share some set of lengths of cycles. For S(3), the elements are 123 -> 123, 132, 213, 231, 312, 321. Its conjugacy classes are
13: {123}
1*2: {132, 213, 321}
3: {231, 312}

The elements of A(n) are all even permutations, those with an even number of even-length cycles. Thus, A(3) has elements 123, 231, 312. Some of the S(n) conjugacy classes get split in two, those with all odd cycle lengths that do not get repeated. Thus, the 3 splits into {231} and {312}.

The remaining coset of A(n) in S(n) is the odd permutations, those with an odd number of even-length cycles. The group (even permutations, odd permutations) is, of course, Z(2).

The representation theory of the symmetric group is very elegant, with the irreducible representations or irreps being labeled with sets of cycle lengths, just like the classes. The character values are all integers.

Going from a symmetric group to its alternating group, the irreps with length sets that are dual to each other get merged, while the irreps with self-dual length sets get split. Duality of the length sets is defined as follows:

Order the lengths from largest to smallest, then make (length) boxes for each one in a 2D pattern. Find out how many boxes there are in the other direction, and that gives the dual. Example:

The length set 3,2,1,1 gives
***
**
*
*
Its dual is 4,2,1

For even permutations, duality leaves the character value's sign unchanged, while for odd permutations, duality reverses the sign. In the self-dual case, odd permutations get zero character.

The irrep (n) for S(n) is the identity irrep, while the irrep (1n) is the parity irrep; it is 1 for even permutations and -1 for odd permutations.

For S(3), the 13 and 3 irreps are a dual set, while the 1*2 irrep is self-dual. Its character table is
[itex]\begin{matrix} & 3 & 1 \cdot 2 & 1^3 \\ 1^3 & 1 & 2 & 1 \\ 1 \cdot 2 & 1 & 0 & -1 \\ 3 & 1 & -1 & 1 \end{matrix}[/itex]
Rows: classes, columns: irreps

For A(3), we get the character table
[itex]\begin{matrix} & (3,1^3) & (1 \cdot 2)_1 & (1 \cdot 2)_2 \\
1^3 & 1 & 1 & 1 \\ 3_1 & 1 & \frac12(-1 + \sqrt{-3}) & \frac12(-1 - \sqrt{-3}) \\ 3_2 & 1 & \frac12(-1 - \sqrt{-3}) & \frac12(-1 + \sqrt{-3}) \end{matrix}[/itex]
The sqrt(-3) is typical of splitting.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
 
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The importance of symmetric groups lies in the fact, that every finite group can be considered a subgroup of a symmetric group: Given a finite group ##G=\{\,a_1,\ldots,a_n\,\}## then the functions ##L_i\, : \,a_j \longmapsto a_ia_j## are permutations of ##\{\,1,\ldots,n\,\}##.
 

Related to What Are Symmetric Groups and Their Mathematical Significance?

What is a symmetric group?

A symmetric group is a mathematical concept that refers to a group of permutations, or rearrangements, of a finite set of objects. In simpler terms, it is a group of all possible ways to arrange a set of items in a symmetric manner.

How is a symmetric group represented?

A symmetric group is typically represented by the notation "Sn", where "n" represents the number of objects in the set. For example, if the set has 4 objects, the symmetric group would be denoted as "S4".

What are the basic properties of a symmetric group?

Some basic properties of a symmetric group include closure, associativity, identity element, and inverse element. Closure means that the result of combining any two elements in the group will always be another element in the group. Associativity means that the grouping of elements in an operation does not affect the result. The identity element is an element that when combined with any other element in the group, will result in that same element. The inverse element is an element that when combined with another element, will result in the identity element.

What is the order of a symmetric group?

The order of a symmetric group is the number of elements in the group. For a symmetric group "Sn", the order would be "n!", where "n" is the number of objects in the set. For example, if the set has 4 objects, the order of the symmetric group would be 4!.

What are some real-life applications of symmetric groups?

Symmetric groups have various applications in fields such as cryptography, group theory, and combinatorics. They are also used in computer science for data encryption and coding theory. Additionally, symmetric groups are used in chemistry to describe the symmetry properties of molecules and in physics to study the symmetry of physical systems.

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