Quotient group

Given a mathematical group G and a normal subgroup N of G, the factor group, or quotient group, of G over N, written as G/N, can be thought of as arising from G by "collapsing" the subgroup N to the identity element. Formally, G/N is the set of all the cosets (see under subgroup) of N in G. There is a natural 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. The identity element of G/N is the coset eN, where e is the identity element of G.

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. Its kernel is N.

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.

Trivially, G/G is isomorphic to the group of order 1, and G/{e} is isomorphic to G. Several important properties of factor groups are recorded in the isomorphism theorems.