Editing Normal subgroup (section)

== Normal subgroups, quotient groups and homomorphisms ==

If <math>N</math> is a normal subgroup, we can define a multiplication on cosets as follows: 
<math display="block">\left(a_1 N\right) \left(a_2 N\right) := \left(a_1 a_2\right) N</math>
This relation defines a mapping <math>G/N\times G/N \to G/N</math>. To show that this mapping is well-defined, one needs to prove that the choice of representative elements <math>a_1, a_2</math> does not affect the result. To this end, consider some other representative elements <math>a_1'\in a_1 N, a_2' \in a_2 N</math>. Then there are <math>n_1, n_2\in N</math> such that <math>a_1' = a_1 n_1, a_2' = a_2 n_2</math>. It follows that <math display="block">a_1' a_2' N = a_1 n_1 a_2 n_2 N =a_1 a_2 n_1' n_2 N=a_1 a_2 N</math>where we also used the fact that <math>N</math> is a {{em|normal}} subgroup, and therefore there is <math>n_1'\in N</math> such that <math>n_1 a_2 = a_2 n_1'</math>. 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 <math>G/N.</math> There is a natural [[Group homomorphism|homomorphism]], <math>f : G \to G/N</math>, given by <math>f(a) = a N</math>. This homomorphism maps <math>N</math> into the identity element of <math>G/N</math>, which is the coset <math>e N = N</math>,{{sfn|Hungerford|2003|pp=42–43}} that is, <math>\ker(f) = N</math>.

In general, a group homomorphism, <math>f : G \to H</math> sends subgroups of <math>G</math> to subgroups of <math>H</math>. Also, the preimage of any subgroup of <math>H</math> is a subgroup of <math>G</math>. We call the preimage of the trivial group <math>\{ e \}</math> in <math>H</math> the '''[[Kernel (algebra)|kernel]]''' of the homomorphism and denote it by <math>\ker f</math>. As it turns out, the kernel is always normal and the image of <math>G, f(G)</math>, is always [[isomorphic]] to <math>G / \ker f</math> (the [[first isomorphism theorem]]).{{sfn|Hungerford|2003|p=44}} In fact, this correspondence is a bijection between the set of all quotient groups of <math>G</math>, <math>G/N</math>, and the set of all homomorphic images of <math>G</math> ([[up to]] isomorphism).{{sfn|Robinson|1996|p=20}} It is also easy to see that the kernel of the quotient map, <math>f : G \to G/N</math>, is <math>N</math> itself, so the normal subgroups are precisely the kernels of homomorphisms with [[Domain of a function|domain]] <math>G</math>.{{sfn|Hall|1999|p=27}}