quotiëntgroup

In abstract algebra, a quotiëntgroup, also known as a factor group, is a group formed by taking the set of all distinct left (or right) cosets of a normal subgroup of a given group. Let G be a group and N be a normal subgroup of G. The set of left cosets of N in G is denoted by G/N. This set G/N forms a group under a well-defined binary operation of coset multiplication. The operation is defined as follows: for any two cosets aN and bN in G/N, their product is (aN)(bN) = (ab)N. This operation is associative because the group operation in G is associative. The identity element in G/N is the coset eN, where e is the identity element of G. Every element in G/N has an inverse. The inverse of aN is a⁻¹N, where a⁻¹ is the inverse of a in G.

The existence of a well-defined operation is crucial and relies on the fact that N is a normal subgroup. This means that for any g in G and n in N, gng⁻¹ is also in N. This normality ensures that the choice of representative element for a coset does not affect the resulting coset. The concept of a quotiëntgroup is fundamental in understanding the structure of groups and is closely related to homomorphisms, particularly through the first isomorphism theorem, which states that if φ: G → H is a surjective group homomorphism, then G/ker(φ) is isomorphic to H, where ker(φ) is the kernel of φ, which is always a normal subgroup of G.