Functions
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Chapter 1 Groups, and other algebraic structures (section 8.1)
1.1 Introduction
A mathematical structure is a formal model which capture common properties
or behavior of different sets.
It consists of an abstract set, together with operations and relations
which obey certain rules.
For example the set of tasks to build a house, and the set of all
courses available at CSM have something in common: the notion of prerequisite.
This notion can be formalized using a structure of partially ordered
set.
We will introduce here certain algebraic structures, which are useful
to model arithmetic, as well as to the study of symmetries.
1.2 Permutations
Définition 1
Let A be a set. A permutation of A is a bijection from
A to A.
Let S_{A} be the set of all permutations of A.
Let S_{n} be the set of all permutations of {1,... n}.
A permutation sigma of S_{n} can be written in array
form as follow:
meaning that sigma(1)=2, sigma(2)=1, sigma(3)=4, and so
on.
Exercice 1
Write all permutations in S_{1}, S_{2}, and S_{3}.
What is the size of S_{n}?
1.2.1 Operation on S_{n}
S_{n} as a set is not very interesting.
Can we define operations on S_{n}?
Proposition 1
The composition ¤is an operation of S_{n}.
Indeed, the composition of two permutations is still a permutation.
We also call this operation product of permutations.
Exercice 2
Compute the following:
The permutation above produces a cycle: 1>2>3>4>5>1
Définition 2
By (a,b,c,d), we denote the permutation sigma such that sigma(a)=b,
sigma(b)=c, sigma(c)=d, sigma(d)=a, and sigma(x)=x
for any other element x.
Such a permutation is called a cycle.
A cycle (i,j) of length two is called a transposition.
A transposition (i,i+1) is called elementary transposition.
Exercice 3
Write (3,4,2,1) and (2,1,3,4) in array notation.
Write the following permutations in array notation:
Proposition 2
The product of two disjoint cycles is commutative.
This is not the case for nondisjoint cycles.
Exercice 4
Are the following permutations cycles?
Problem 1
So there are permutations that are not cycles.
Would it be possible to write them using only cycles?
Théorème 1
Any permutation can be written as a product of disjoint cycles.
This product is unique, up to the order.
Exercice 5
Compute the following permutations

(
( 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
5 
10 
2 
1 
6 
4 
8 
3 
7 
9 

)
) 

= 
( 

) 
Writting a permutation in cycle notation is a good way to compute
powers of this permutation.
1.2.2 Generators of S_{n}
If you have a list, you can use a permutation to change its order.
Well, S_{n} is pretty big, so this makes quite a lot of possible
operations.
However, some operations can be obtained by combining other operations.
Could we try to reduce the size of S_{n}?
Définition 3
A subset G of S_{n} generates S_{n} if any permutation
can be written as a product of elements of G.
Problem 2
Find a set of generators of S_{n} as small as possible.
Problem 3
Does the set of all cycles generate S_{n}?
Problem 4
How many cycles are there ?
Can you do better?
Problem 5
Does the set of all transpositions generate S_{n}?
How many transpositions are there?
Can you do better?
Problem 6
Does the set of all elementary transposition generate S_{n}?
How many elementary transpositions are there ?
By the way, how many elementary transpositions at most do you need
to express a permutation?
Can you do better than n1 generators?
1.3 Groups
1.3.1 Definitions and examples
The notation [A,¤] denote a set A together with an operation
¤."
Problem 1
Is there anything in common between [S_{n},¤] and [Z +].
Définition 4
Let A be a set with an operation ¤
What are the properties of [Z,+]?

+ associative;
 +is commutative;
 0 is an identity element;
 If x is an integer, thenx is an inverse for x.
What are the properties of [R^{*},·]?
Those structures have many common properties.
Définition 5
[G,¤] is a group
if G is a non empty set, and ¤is
a binary operation on G such that:

¤ is associative
 an identity element exists (in G).
 each element of G has an inverse (in G) with respect to ¤.
If the operation ¤is commutative, G is called a commutative
group
.
Exercice 6
Which of the following are groups ?

[Z,+]
 [N,+]
 [Z,]
 [{ true,false},/\}
 [R,+]
 [R,·]
 [S_{n},¤]
 [nZ,+]
What's the point ?

Many structures, coming in particular from arithmetics, are groups.
 If you prove a property for groups in general, you have proved this
property for all of those structures.
 Group theory is a vast field, and there are hundreds of theorems about
groups waiting for you to apply them on your favorite structure.
 Basic tool to study the Rubicks Cube.
1.3.2 Just some examples of results and questions about groups
Théorème 2
The identity of a group is unique.
The inverse of an element is unique.
The equations a¤ x=b and x¤ a=b have a unique solution.
The inverse of a product can be computed by (x¤ y)^{1}=y^{1}¤ x^{1}.
Définition 6
Let [G,¤] be a group, and Aincluded in or equal to G.
Then [A,¤] is a subgroup of [G,¤] if [A,¤]
itself is a group.
Théorème 3
The size of a subgroup divides the size of the group.
Problem 2
Common problems about a group:

Find a minimal set of generators
 Express an element of the group as product of generators
 Efficient computations
 Find all subgroups
 Are two groups structurally the same (are they isomorphic?)
For playing with groups, there is a very nice program:
Functions
Class notes
Groups and other algebraic structures /
CSM MACS358 /
Nicolas M. Thiéry
Last modified: Fri Jan 19 13:12:22 2007