the index of H in G

The concept of set is fundamental in groups , so we briefly some notions about sets , functions , and number theory . The student study this subjects , so we will briefly summarizes of it .
Definition 1 – 1 .
If A is an arbitrary set , then the set whose elements are all the subsets of A is known as the POWER SET of A and denoted by P (A) :
P (A) = { B ?B ?A } .
Example:
If A = { a , b } , find P ( A ) .
Solution : H . W .

Notes:
If A is a finite set with n elements , then P ( A ) having 2n element s .
P ( A ) ? ?
Definition 1 – 2 .
The CARTESIAN PRODUCT of two nonempty sets A and B , denoted by A × B , is the set
A ×B = { (a,b) ?a ? A and b ? B }
Definition 1 – 3 .
Let A be a nonempty set , a subset R of the Cartesian product A × A is called RELATION .

1
Definition 1 – 4 .
A relation R in a set A is said to be an EQUIVALENCE RELATION in A provided it satisfied the properties ,
reflexive: aRa , for each a ? A ,
symmetric : if aRb for some a, b ? , then b R a ,
transitive : if aRb and bRc for some a , b , c ? A , then aRc .
Equivalence relations are denoted by ~ .
Example .
Let A be the set of all lines in a plane . For a , b ? A let a ? ? b means that a is parallel to b .If we agree that every line is parallel to itself , then ? ? is equivalence relations .
Definition 1 – 5 .
Let Z be the set of integer numbers , and let n be a fixed positive integer . For any a , b ? Z , a is said to be congruent to be (modulo n), written
a ? b (mod n),
if and only if the deference a – b is divisible by n.
that is mean , a ? b (mod n), if and only if a – b = kn for some integer n.
Example.
If n =5, we note
2 ? 7 ( mod 5)
-1 ? 4 ( mod 5)
Definition 1-6 .
Let ~ be an equivalence relations on A , and let a ? A . The set of all elements in A which are equivalent to a is called the EQUIVALENCE CLASS of a under ~ and is denoted by [ a ] :
[ a ] = { b ?A? b ~ a } .
2
Theorem 1-1.
Let ~ be an equivalence relation in the set A . Then,
1)for each a ? A , [ a ] ? ? ,
2) if b ? [ a ] , then [ a ] = [ b ] ,
3) for any a , b ? A , with [a ] ? [ b ] , [ a ] ? [ b ] =? ,
4) U { [ a ] | a ? A } = A .
Proof .
~ is equivalence relation , so it is reflexive .
a ~ a , then a ? [ a].
Hence [a] ? ?
Let b? [a],
Hence b ~ a .
Since ~ is symmetric , then a ~ b.
Suppose that x ? [a], then x~a .
Since ~ is transitive , then x ~ b.
Hence x ? [b] .
So [a] ? [b].
Similarly [b] ? [a]
suppose that [a] ? [b] ? ?,
So there exist c ? [a] ? [b].
By 2 [a] = [c] = [b] , which is contradiction .
So [a] ? [b] = ?.
Since [a] ? A , then U {[a]|a ? A } ? A.
Since for each a ? A, then a ? [a].
Hence A ? U{[a] | a? A }.
So U{[a] |a? A } = A
Definition 1 - 7 .

A partition of a set A is a family {Ai} of nonempty subsets of A with the properties
If Ai ? Aj , then Ai ? Aj = ? (pairwise disjoint),
U Ai = A.
Example
If Zeis the set of even integers,
andZo is the set of odd integers,
thenZeU Zo = Z,
andZe ? Zo = ?.
Hence {Ze ,Zo} is partition of Z.
Example.
If A = { 0 ,1, 2 , . . . , 24 } , n= 5 , then
{ [ 0 ] , [ 1 ] , [ 2 ] , [ 3 ] , [ 4 ] } is partition of A.

Theorem 1-2 .
If {Ai} is a partition of the set A, then there is an equivalence . relation in A whose equivalence classes are precisely the . sets Ai.