Normal subgroup

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In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. 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.

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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−1N.
    • 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.

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