Normal Subgroup - Definitions

Definitions

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. We write

For any subgroup, the following conditions are equivalent to normality. Therefore any one of them may be taken as the definition:

• For all g in G, gNg−1 ⊆ N.
• For all g in G, gNg−1 = N.
• The sets of left and right cosets of N in G coincide.
• For all g in G, gN = Ng.
• N is a union of conjugacy classes of G.
• There is some homomorphism on G for which N is the kernel.

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.