Normal divisor

In group theory, a branch of mathematics, a normal subgroup, also known as invariant subgroup, or normal divisor, is a (proper or improper) subgroup H of the group G that is invariant under conjugation by all elements of G.

Two elements, a&prime; and a,  of  G are said to be conjugate by g &isin; G,  if a&prime;  = g a g&minus;1. Clearly, a = g&minus;1 a&prime; g, so that conjugation is symmetric; a and a&prime; are conjugate partners.

If for all h &isin; H and all g &isin; G it holds that:  g h g&minus;1 &isin; H,  then H is a normal subgroup of G, (also expressed as "H is invariant in G"). That is, with h in H all conjugate partners of h are also in H.

Equivalent definitions
A subgroup H of a group G is termed normal if the following equivalent conditions are satisfied:


 * 1) Given any $$h \in H$$ and $$g \in G$$, we have $$ghg^{-1} \in H$$
 * 2) H occurs as the kernel of a homomorphism from G. In other words, there is a homomorphism $$\phi: G \to K$$ such that the inverse image of the identity element of K is H.
 * 3) Every inner automorphism of G sends H to within itself
 * 4) Every inner automorphism of G restricts to an automorphism of H
 * 5) The left cosets and right cosets of H are always equal: $$x H = H x$$. (This is often expressed as: "H is simultaneously left- and right-invariant").

Klein's Vierergruppe in S4
The set of all permutations of 4 elements forms the symmetric group S4, which is of order of 4! = 24. The group of the following four permutations is a subgroup and has the structure of Felix Klein's Vierergruppe:
 * V4 &equiv; {(1), (12)(34), (13)(24), (14)(23)}

It is easily verified that V4 is a normal subgroup of S4. [Conjugation preserves the cycle structure (..)(..) and V4 contains all elements with this structure.]

All subgroups in Abelian groups
In an Abelian group, every subgroup is normal. This is because if $$G$$ is an Abelian group, and $$g,h \in G$$, then $$ghg^{-1} = h$$.

More generally, any subgroup inside the center of a group is normal.

It is not, however, true that if every subgroup of a group is normal, then the group must be Abelian. A counterexample is the quaternion group.

All characteristic subgroups
A characteristic subgroup of a group is a subgroup which is invariant under all automorphisms of the whole group. Characteristic subgroups are normal, because normality requires invariance only under inner automorphisms, which are a particular kind of automorphism.

In particular, subgroups like the center, the commutator subgroup, the Frattini subgroup are examples of characteristic, and hence normal, subgroups.

A smallest counterexample
The smallest example of a non-normal subgroup is a subgroup of order two in the symmetric group on three elements. Explicitly, we can take the cyclic subgroup of order two generated by the 2-cycle $$(12)$$ in the symmetric group of permutations on symbols $$1,2,3$$.

Properties
The intersection of any family of normal subgroups is again a normal subgroup. We can therefore define the normal subgroup generated by a subset S of a group G to be the intersection of all normal subgroups of G containing S.

Quotient group
The quotient group of a group G by a normal subgroup N is defined as the set of (left or right) cosets:


 * $$G/N = \{ Nx : x \in G \} \, $$

with the the group operations


 * $$ Nx . Ny = N (xy) \, $$
 * $$ (Nx)^{-1} = N x^{-1} \, $$

and the coset $$N = N1$$ as identity element. It is easy to check that these define a group structure on the set of cosets and that the quotient map $$q_N : x \mapsto N x$$ is a group homomorphism. Because of this property N is sometimes called a normal divisor of G.

First Isomorphism Theorem
The First Isomorphism Theorem for groups states that if $$f : G \rightarrow H$$ is a group homomorphism then the kernel of f, say K, is a normal subgroup of G, and the map f factors through the quotient map and an injective homomorphism i:


 * $$G \stackrel{q_K}{\longrightarrow} G/K \stackrel {i}{\longrightarrow} H . \, $$