Factor group

Given a mathematical group G and a normal subgroup N of G, the factor group of G over N is the set of all the cosets (see under subgroup) of N in G; it is denoted by G/N. Being N normal in G allows one to give in a natural way a group structure on G/N: if aN and bN are elements of G/N, the product aN * bN is by definition equal to (ab)N, and by the normality of N this definition is well-posed.

There is a "natural" surjective group homomorphism π : G -> G/N, sending each element g of G in the coset of N to which it belongs, that is: π(g) = gN. The application π is sometimes called canonical projection.

When G/N is finite, its order is equal to [G:N], the index of N in G.

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