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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).
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:
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.
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 well-defined (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−1N. 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 S1, 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 non-zero 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.
Every group is isomorphic to a quotient of a free group.