Quotient group

In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively "collapses" the normal subgroup N to the identity element. The quotient group is written G/N and is usually spoken in English as G mod N (mod is short for modulo).
Contents 
The product of subsets of a group
In the following discussion, we will use a binary operation on the subsets of G: if two subsets S and T of G are given, we define their product as:
 <math>ST = \{ st : s \isin S {\rm~and~} t \isin T \}<math>
This operation is associative and has identity element {e}, where e is the identity element of G. Thus, the set of all subsets of G forms a monoid under this operation.
A subgroup N of a group G is normal if and only if the coset equality aN = Na holds for all a in G. In terms of the binary operation on subsets defined above, a normal subgroup of G is a subgroup that commutes with every subset of G.
Definition
We define the set G/N to be the set of all left cosets of N in G, i.e.
 <math>G/N = \{ aN : a \isin G \}<math>
The group operation on G/N is the product of subsets defined above. In other words, for each aN and bN in G/N, the product of aN and bN is (aN)(bN). For this operation to be closed, we must show that (aN)(bN) really is a left coset:
 (aN)(bN) = a(Nb)N = a(bN)N = (ab)NN = (ab)N
Note that we have already used the normality of N in this equation. Also note that because of the normality of N, we could have chosen to define G/N as the set of right cosets of N in G. Also note that because the operation is derived from the product of subsets of G, the operation is welldefined (does not depend on the particular choice of representatives), associative and has identity element N.
The inverse of an element aN of G/N is a^{−1}N. This completes the proof that G/N is a group.
Examples
Consider the group of integers Z (under addition) and the subgroup 2Z consisting of all even integers. This is a normal subgroup, because Z is abelian. There are only two cosets, the set of even integers and the set of odd integers, and Z/2Z is the cyclic group with two elements.
As another abelian example, consider the group of real numbers R (again under addition) and the subgroup Z of integers. The cosets of Z in R are all sets of the form a + Z, with 0 ≤ a < 1 a real number. Adding such cosets is done by adding the corresponding real numbers, and subtracting 1 if the result is greater than or equal to 1. The factor group R/Z is isomorphic to S^{1}, the group of complex numbers of absolute value 1 under multiplication. An isomorphism is given by f(a + Z) = exp(2πia) (see Euler's identity).
If G is the group of invertible 3×3 real matrices, and N is the subgroup of 3×3 real matrices with determinant 1, then N is normal in G (since it is the kernel of the determinant homomorphism), and G/N is isomorphic to the multiplicative group of nonzero real numbers.
Properties
Trivially, G/G is isomorphic to the trivial group (the group with one element), and G/{e} is isomorphic to G.
The order of G/N is by definition equal to [G : N], the index of N in G. If G is finite, the index is also equal to the order of G divided by the order of N. Note that G/N may be finite, although both G and N are infinite (e.g. Z/2Z).
There is a "natural" surjective group homomorphism π : G → G/N, sending each element g of G to the coset of N to which g belongs, that is: π(g) = gN. The mapping π is sometimes called the canonical projection of G onto G/N. Its kernel is N.
There is a bijective correspondence between the subgroups of G that contain N and the subgroups of G/N; if H is a subgroup of G containing N, then the corresponding subgroup of G/N is π(H). This correspondence holds for normal subgroups of G and G/N as well, and is formalized in the lattice theorem.
Several important properties of quotient groups are recorded in the fundamental theorem on homomorphisms and the isomorphism theorems.
If G is abelian, nilpotent or solvable, then so is G/N.
If G is cyclic or finitely generated, then so is G/N.
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