Factor group

Given a mathematical group G and a normal subgroup N of G, the factor group, or quotient group, of G over N can be thought of as arising from G by "collapsing" the subgroup N to the identity element. It is written as G/N.

Definition

Formally, G/N is the set of all left cosets of N in G, i.e. G/N = {aN : a in G}. Generally, if two subsets S and T of G are given, we define their product as ST = {st : s in S and t in T}. This operation on subsets of G is associative, and if N is normal, then it turns the set G/N into a group: (aN)(bN) = (aN)(Nb) = a((NN)b) = a(Nb) = a(bN) = (ab)N which establishes closure. From the same calculation, it follows that eN=N is the neutral element of G/N and that a^-1N is the inverse of aN.

Examples

Consider the group of integers Z (using addition as operation) and the subgroup 2Z consisting of all even integers. This is a normal subgroup because Z is abelian. There are only two cosets, the even and the odd numbers, and Z / 2Z is therefore isomorphic to the cyclic group with two elements.

As another abelian example, consider the group of real numbers R (again with addition) and the subgroup Z of integers. The cosets of Z in R are 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 using multiplication as operation. An isomorphism is given by f(a + Z) = exp(a 2πi) (see Euler's identity).

If G is the group of invertible 3-by-3 real matrices, and N is the subgroup of 3-by-3 real matrices with determinant 1, then N is normal in G (since it is the kernel of the group homomorphism det), and G/N is isomorphic to the multiplicative group of non-zero real numbers.

Properties

Trivially, G/G is isomorphic to the group of order 1, and G/{e} is isomorphic to G.

When G/N is finite, its order is equal to [G:N], the index of N in G. If G is finite, this is also equal to the order of G divided by the order of N; this may explain the notation.

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 application π is sometimes called canonical projection. 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 factor 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 group of the form F/N, where F is a free group and N is a normal subgroup of F.