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In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup H of a group G is normal in G if and only if gH = Hg for all g in G, i.e., the sets of left and right cosets coincide. Normal subgroups (and only normal subgroups) can be used to construct quotient groups from a given group.
Évariste Galois was the first to realize the importance of the existence of normal subgroups.^{[1]}
A subgroup N of a group G is called a normal subgroup if it is invariant under conjugation; that is, for each element n in N and each g in G, the element gng^{−1} is still in N.^{[2]} We write
For any subgroup, the following conditions are equivalent to normality. Therefore any one of them may be taken as the definition:
The last condition accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images,^{[3]} a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.
The normal subgroups of a group G form a lattice under subset inclusion with least element {e} and greatest element G. Given two normal subgroups N and M in G, meet is defined as
and join is defined as
The lattice is complete and modular.
If N is normal subgroup, we can define a multiplication on cosets by
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.
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