Normal subgroup

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.

A subgroup, N, of a group, G, is called a normal subgroup if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is still in N:

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, 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

...
Wikipedia