Symmetric Group

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date: 2025-01-03

Let denote the set of all bijections on the set if define to be the bijection

This gives us a binary operation on which is associative, the identity permutation is such that and is such that for every the inverse bijection denoted by such that
is a group is called the symmetric group of degree n

Definition

A cycle is a string of positive integers say, which represents the permutation such that and fixes all other integers

Example

( is the only non-abelion group of order 6)

Exercise

Show that group of order has only 1 group up to isomorphism that is

Remark

There are 2 groups of order 4

Remark

A cycle with contains integers is said to be of length A - Cycle is of order ..,

Definition

Two cycles in are called disjoint if they have no integer in common.

Remark

If and are two disjoint cycles in then and commute, that is =

Claim

Every can be written uniquely as a product of disjoint cycles

example: let

Remark

2-cycles are also called transpositions

Remark

Every cycle can be written as a product of 2 cycles >

Exercise

If a cycle can be expressed as a product of an even number of transpositions, it is always an even permutation. Similarly, if it is expressed as a product of an odd number of transpositions, all the transpositions will be odd.

Let be indeterminates

Let

Definition

A permutation is said to be even if and is said to be odd if Sign of a permutation denoted by is +1 if is even, -1 if is odd so

The map is the sign of satisfies the following satisfies Let there be factors such that and

Now, has exactly factors of the form Bring out a factor of we have has all factor of the form ,

Remark

A cycle of length is an even permutation IFF is odd. This is because can be written as the product of odd permutation that is k-1 permutation

Proposition

If a transposition, then

Let if then where interchanges 1, and 2,

using that is homomorphism

Proposition

if if is an -cycle then

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