stylistic:style_transfer, information:extraction, linguistic:voice_change, information:classification, structural:expansion
Normal subgroups, quotient groups and homomorphisms

If N {\displaystyle N} is a normal subgroup, we can define a multiplication on cosets as follows:

This relation defines a mapping G / N × G / N → G / N {\displaystyle G/N\times G/N\to G/N}. To show that this mapping is well-defined, one needs to prove that the choice of representative elements a 1, a 2 {\displaystyle a_{1},a_{2}} does not affect the result. To this end, consider some other representative elements a 1 ′ ∈ a 1 N, a 2 ′ ∈ a 2 N {\displaystyle a_{1}'\in a_{1}N,a_{2}'\in a_{2}N}. Then there are n 1, n 2 ∈ N {\displaystyle n_{1},n_{2}\in N} such that a 1 ′ = a 1 n 1, a 2 ′ = a 2 n 2 {\displaystyle a_{1}'=a_{1}n_{1},a_{2}'=a_{2}n_{2}}. It follows that

where we also used the fact that N {\displaystyle N} is a normal subgroup, and therefore there is n 1 ′ ∈ N {\displaystyle n_{1}'\in N} such that n 1 a 2 = a 2 n 1 ′ {\displaystyle n_{1}a_{2}=a_{2}n_{1}'}. This proves that this product is a well-defined mapping between cosets.

With this operation, the set of cosets is itself a group, called the quotient group and denoted with G / N. {\displaystyle G/N.} There is a natural homomorphism, f: G → G / N {\displaystyle f:G\to G/N}, given by f (a) = a N {\displaystyle f(a)=aN}. This homomorphism maps N {\displaystyle N} into the identity element of G / N {\displaystyle G/N}, which is the coset e N = N {\displaystyle eN=N}, that is, ker (f) = N {\displaystyle \ker(f)=N}.

In general, a group homomorphism, f: G → H {\displaystyle f:G\to H} sends subgroups of G {\displaystyle G} to subgroups of H {\displaystyle H}. Also, the preimage of any subgroup of H {\displaystyle H} is a subgroup of G {\displaystyle G}. We call the preimage of the trivial group { e } {\displaystyle \{e\}} in H {\displaystyle H} the kernel of the homomorphism and denote it by ker f {\displaystyle \ker f}. As it turns out, the kernel is always normal and the image of G, f (G) {\displaystyle G,f(G)}, is always isomorphic to G / ker f {\displaystyle G/\ker f} (the first isomorphism theorem). In fact, this correspondence is a bijection between the set of all quotient groups of G {\displaystyle G}, G / N {\displaystyle G/N}, and the set of all homomorphic images of G {\displaystyle G} (up to isomorphism). It is also easy to see that the kernel of the quotient map, f: G → G / N {\displaystyle f:G\to G/N}, is N {\displaystyle N} itself, so the normal subgroups are precisely the kernels of homomorphisms with domain G {\displaystyle G}.