Normal Subgroups and Homomorphisms
If N is normal subgroup, we can define a multiplication on cosets by
- (a1N)(a2N) := (a1a2)N.
This turns the set of cosets into a group called the quotient group G/N. There is a natural homomorphism f: G → G/N given by f(a) = aN. The image f(N) consists only of the identity element of G/N, the coset eN = N.
In general, a group homomorphism f: G → H sends subgroups of G to subgroups of H. Also, the preimage of any subgroup of H is a subgroup of G. We call the preimage of the trivial group {e} in H the kernel of the homomorphism and denote it by ker(f). As it turns out, the kernel is always normal and the image f(G) of G is always isomorphic to G/ker(f) (the first isomorphism theorem). In fact, this correspondence is a bijection between the set of all quotient groups G/N of G and the set of all homomorphic images of G (up to isomorphism). It is also easy to see that the kernel of the quotient map, f: G → G/N, is N itself, so we have shown that the normal subgroups are precisely the kernels of homomorphisms with domain G.
Read more about this topic: Normal Subgroup
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