Quotient Group - recall Weyl Group in SU(3)

Normal Subgroup and Equivalence

Let G be a group and N a subgroup of G. Then, N is called a normal subgroup of G if for each element g of G and each element n of N, the element gng^-1 belongs to N. Note that if G is commutative, then every subgroup of G is automatically normal because gng^-1 = n. 

If G and H are groups and Φ : G --> H is a homomorphism, then ker Φ is a normal subgroup of G. Let e₂ denote the identity element of H and suppose n is an element of ket Φ (i.e., Φ(n) = e₂). Then Φ(gng^-1) = e₂. Thus, ket Φ is a normal subgroup of G. 

Suppose G is a group and N is a normal subgroup of G. Define two elements g and h to be equivalent if gh^-1 ∈ N.

1. If g₁ is equivalent to g₂ and h₁ is equivalent to h₂, then g₁h₁ is equivalent to g₂h₂.

2. If g₁ is equivalent to g₂, then g₁-1 is equivalent to g₂-1 .

Quotient Group

If g is any element of G, let [g] denote the equivalence class containing G; that is [g] is the subset of G consisting of all elements equivalent to g (including g itself). 

Let G be a group and N a normal subgroup of G. The quotient group G/N is the set of all equivalence classes in G, with the product operation defined by [g][h] = [gh]. The elements of G/N are equivalence classes. The group product is defined by choosing one element g out of the first equivalence class, choosing one element h out of the second equivalence class, and then define the product to be the equivalence class containing gh. 

The idea behind the quotient group construction is that we make a new group out of G by setting every element of N equal to the identity. This then forces ng to be equal to g ∈ G and n ∈ N. However, ng and n are equivalent, since (ng)g^-1 = n ∈ N. So setting elements of N equal to the identity forces elements that are equivalent to be equal. The condition that N be a normal subgroup guarantees that we still have defined group operations after setting equivalent elements equal to each other. 

If G is a group and N is a normal subgroup, then there is a homomorphism q of G into the quotient group G/N given by q(g) = [g]. Follows from the definition of the product operation on G/N that q is indeed a homomorphism and it clearly maps G onto G/N. For Φ : G --> H is homomorphism, we have seen that ker Φ is a normal subgroup of G. If Φ maps G onto H, then it can be shown that H is homomorphic to the quotient group G/ker Φ. 

Note: If G is a matrix Lie group, then G/N may not be. Even if G/N happens to be a matrix Lie group, there is no canonical procedure for finding a matrix representation of it. 

Examples

1. Group of integers modulo n. In this case, G = Z and N = nZ (set of integer multiples of n). To form the quotient group, we say that two elements of Z are equivalent if their difference is in N. Thus, the equivalence class of an integer i is the set of all integers that are equal modulo n to i.

2. Taking G = SL(n;C) and taking N to be the set of elements of SL(n;C) that are multiples of the identity. The elements of N are the matrices of the form e^2πik/nI, k = 0, 1, ...., n-1. This is a normal subgroup of SL(n;C) because each element of N is a multiple of the identity and, thus, for any A ∈ SL(n;C), we have A(e^2πik/nI)A^-1 = AA^-1(e^2πik/nI) = e^2πik/nI. The quotient group SL(n;C)/N is customarily denoted PSL(n;C), where P stands for "projective". It can shown that PSL(n;C) is a simple group for all n ≥ 2; that is, PSL(n;C) has no normal subgroups other than {I} and PSL(n;C) itself.