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 gng⁻¹ is still in N. The statement N is a normal subgroup of G is written:

<math>N\triangleleft G</math>.

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, gNg⁻¹ ⊆ N.
2. For all g in G, gNg⁻¹ = 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.
6. There is some homomorphism on G for which N is the kernel.

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(Ng⁻¹) = g(g⁻¹N) = (gg⁻¹)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 <math> N_1 N_2 =\{ n_1 n_2 \,|\,n_1 \in N_1, \,n_2 \in N_2 \}</math> which is a group because one of the factors (N₁ or N₂) in the group product is a normal subgroup.

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

Contents 1 Example 2 Normal subgroups and homomorphisms 3 Attributes of normality 4 References 5 See also

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 : G → G/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: G → H 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: G → G/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!.
• Even more generally, if p is the smallest prime dividing the order of G, then every subgroup of index p is normal.

References

• I. N. Herstein, Topics in algebra. Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp.
• David S. Dummit; Richard M. Foote, Abstract algebra. Prentice Hall, Inc., Englewood Cliffs, NJ, 1991. xiv+658 pp. ISBN 0-13-004771-6

See also

Operations taking subgroups to subgroups:

• normalizer
• normal closure
• normal core

Subgroup properties stronger than normality:

• characteristic subgroup
• fully characteristic subgroup

Subgroup properties weaker than normality:

• subnormal subgroup
• ascendant subgroup
• descendant subgroup
• serial subgroup
• quasinormal subgroup
• seminormal subgroup
• conjugate permutable subgroup
• modular subgroup
• pronormal subgroup
• paranormal subgroup
• polynormal subgroup
• c normal subgroup

Subgroup properties complementary (or opposite) to normality:

• malnormal subgroup
• contranormal subgroup
• abnormal subgroup
• self-normalizing subgroup

Related notions in algebra:

• ideal (ring theory)cs:Normální podgrupa

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