Normal subgroup

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In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written:

.

There are a number of conditions which are equivalent to requiring that a subgroup N be normal in G. Any one of them may be taken as the definition:

  1. For all g in G, g−1NgN.
  2. For all g in G, g−1Ng = N.
  3. The sets of left and right cosets of N in G coincide.
  4. For each g in G, gN = Ng.
  5. N is a union of conjugacy classes of G.

Note that condition (1) is logically weaker than condition (2), and condition (3) is logically weaker than condition (4). For this reason, conditions (1) and (3) are often used to prove that N is normal in G, while conditions (2) and (4) are used to prove consequences of the normality of N in G.

{e} and G are always normal subgroups of G. If these are the only ones, then G is said to be simple.

All subgroups N of an abelian group G are normal, because g−1(Ng) = g−1(gN) = (g−1g)N = N. A group that is not Abelian but for which every subgroup is normal is termed a Hamiltonian group.

The normal subgroups of any group G form a lattice under inclusion. The minimum and maximum elements are {e} and G, the greatest lower bound of two normal subgroups N1 and N2 is their intersection and their least upper bound is their group product, defined as the set (in G) of products which is a group because one of the factors (N1 or N2) in the group product is a normal subgroup.

Galois was the first to realize the importance of the existence of normal subgroups.

Example

The translation group in any dimension is a normal subgroup of the Euclidean group; for example in 3D:

  • rotating, translating, and rotating back results in only translation; also reflecting, translating, and reflecting again results in only translation (a translation seen in a mirror looks like a translation, with a reflected translation vector)
  • the translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances

Normal subgroups and homomorphisms

Normal subgroups are of relevance because if N is normal, then the quotient group G/N may be formed: if N is normal, we can define a multiplication on cosets by

(a1N)(a2N) := (a1a2)N

This turns the set of cosets into a group called the quotient group G/N. There is a natural homomorphism f : GG/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: GH 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). 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: GG/N, is N itself, so we have shown that the normal subgroups are precisely the kernels of homomorphisms with domain G.

Attributes of normality

  • The intersection of a family of normal subgroups is normal
  • The subgroup generated by a family of normal subgroups is normal
  • Normality is preserved upon surjective homomorphisms, and is also preserved upon taking inverse images.
  • Normality is preserved on taking direct products
  • A normal subgroup of a normal subgroup need not be normal. That is, normality is not a transitive property. However, a characteristic subgroup of a normal subgroup is normal. Also, a normal subgroup of a central factor is normal. In particular, a normal subgroup of a direct factor is normal.
  • Every subgroup of index 2 is normal. More generally, a subgroup H of finite index n in G contains a subgroup K normal in G and of index dividing n!.

See also

cs:Normální podgrupa de:Normalteiler es:Subgrupo normal fr:Sous-groupe distingué it:Sottogruppo normale he:תת חבורה נורמלית pl:Dzielnik normalny