In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they 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, for each element, n, in N and each g in G,...
more
In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. Normal subgroups are important because they 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, for each element, n, in N and each g in G, the element gng is still in N. We write
For any subgroup, the following conditions are equivalent to normality. Therefore any one of them may be taken as the definition:
*These are logically stronger than the conditions above them and are not necessary for N to be a subgroup. They are properties of the subgroup.
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...
less
