Normal subgroup

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In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup H of a group G is normal in G if and only if gH = Hg for all g in G. The definition of normal subgroup implies that the sets of left and right cosets coincide. In fact, a seemingly weaker condition that the sets of left and right cosets coincide also implies that the subgroup H of a group G is normal in G. Normal subgroups (and only 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.^[1]

Definitions

A subgroup N of a group G is called a normal subgroup if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is always in N:^[2]

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

• Any two elements commute regarding the normal subgroup membership relation: ∀g, h ∈ G, gh ∈ N ⇔ hg ∈ N.
• The image of conjugation of N by any element of G is a subset of N: ∀g ∈ G, gNg⁻¹ ⊆ N.^[3]
• The image of conjugation of N by any element of G is N: ∀g ∈ G, gNg⁻¹ = N.^[3]
• ∀g ∈ G, gN = Ng.^[3]
• The sets of left and right cosets of N in G coincide.^[3]
• The product of an element of the left coset of N with respect to g and an element of the left coset of N with respect to h is an element of the left coset of N with respect to gh: ∀x, y, g, h ∈ G, x ∈ gN and y ∈ hN ⇒ xy ∈ (gh)N.
• N is a union of conjugacy classes of G: N = ⋃_g∈N Cl(g).^[1]
• N is preserved by inner automorphisms.^[4]
• There is some homomorphism on G for which N is the kernel: ∃φ ∈ Hom(G) ∣ ker φ = N.^[1]

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,^[5] 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.

Examples

• The subgroup {e} consisting of just the identity element of G and G itself are always normal subgroups of G. The former is called the trivial subgroup, and if these are the only normal subgroups, then G is said to be simple.^[6]
• The center of a group is a normal subgroup.^[7]
• The commutator subgroup is a normal subgroup.^[8]
• More generally, any characteristic subgroup is normal, since conjugation is always an automorphism.^[9]
• Every subgroup N of an abelian group G is normal, because gN = Ng. A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group.^[10]
• The translation group is a normal subgroup of the Euclidean group in any dimension.^[11]
• In the Rubik's Cube group, the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal.^[12]

Properties

• Normality is preserved upon surjective homomorphisms, and is also preserved upon taking inverse images.^[13]
• Normality is preserved on taking direct products.^[14]
• If H is a normal subgroup of G, and K is a subgroup of G containing H, then H is a normal subgroup of K.^[15]
• A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8.^[16] However, a characteristic subgroup of a normal subgroup is normal.^[17] A group in which normality is transitive is called a T-group.^[18]
• 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! called the normal core. In particular, if p is the smallest prime dividing the order of G, then every subgroup of index p is normal.^[19]

Normal subgroups and homomorphisms

If N is a normal subgroup, we can define a multiplication on cosets as follows:

With this operation, the set of cosets is itself a group, called the quotient group and denoted 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.^[20]

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 of G, f(G), is always isomorphic to G/ker(f) (the first isomorphism theorem).^[21] In fact, this correspondence is a bijection between the set of all quotient groups of G, G/N, and the set of all homomorphic images of G (up to isomorphism).^[22] It is also easy to see that the kernel of the quotient map, f: G → G/N, is N itself, so the normal subgroups are precisely the kernels of homomorphisms with domain G.^[23]

See also

Notes

References

Further reading

• I. N. Herstein, Topics in algebra. Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp.

External links

• Weisstein, Eric W. "normal subgroup". MathWorld.
• Normal subgroup in Springer's Encyclopedia of Mathematics
• Robert Ash: Group Fundamentals in Abstract Algebra. The Basic Graduate Year
• Timothy Gowers, Normal subgroups and quotient groups
• John Baez, What's a Normal Subgroup?