Normal divider

Normal divisors are special subgroups considered in the mathematical branch of group theory , they are also called normal subgroups.

Their importance lies above all in the fact that they are exactly the kernels of group homomorphisms . These images between groups make it possible to isolate individual aspects of the structure of a group in order to be able to study them more easily in the pure form of the picture group .

The term "... divisor" refers to the fact that a group of factors and each of its normal divisors can be used to form a factor group. These factor groups are homomorphic images of , and each homomorphic image of is isomorphic to such a factor group . ${\ displaystyle G}$${\ displaystyle N}$ ${\ displaystyle G / N}$${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle G / N}$

In the 19th century, the French mathematician Évariste Galois was the first to recognize the importance of the concept of “normal divisors” for studying non-commutative groups . In his theory for solving algebraic equations, the so-called Galois theory , the existence of normal divisors of a group of permutations ( Galois group ) is decisive for the solvability of the equation by radicals .

Sentence and definition

Let it be a subgroup of the group and any element of . The left minor class of after the element of is the subset ${\ displaystyle N}$${\ displaystyle G}$${\ displaystyle g}$${\ displaystyle G}$ ${\ displaystyle gN}$${\ displaystyle N}$${\ displaystyle g}$${\ displaystyle G}$

${\ displaystyle gN: = \ {gn \ mid n \ in N \} \ subseteq G}$.

In the same way, the right minor class of after the element is declared as ${\ displaystyle N}$${\ displaystyle g}$

${\ displaystyle Ng = \ {ng \ mid n \ in N \} \ subseteq G}$.

One can show that the following five statements are pairwise equivalent for a subgroup : ${\ displaystyle N \ subseteq G}$

1. Applies to everyone . (One also says: is invariant under the conjugation with .)${\ displaystyle g \ in G}$${\ displaystyle gNg ^ {- 1} = N}$${\ displaystyle N}$${\ displaystyle g}$
2. For each and every one applies , that is .${\ displaystyle g \ in G}$${\ displaystyle n \ in N}$${\ displaystyle gng ^ {- 1} \ in N}$${\ displaystyle \ forall g \ in G \ colon gNg ^ {- 1} \ subseteq N}$
3. For each the left agrees with the right coset of agreement: .${\ displaystyle g \ in G}$${\ displaystyle N}$${\ displaystyle \ forall g \ in G \ colon gN = Ng}$
4. The set is a union of conjugation classes of the group .${\ displaystyle N}$${\ displaystyle G}$
5. There is a group homomorphism from whose core is.${\ displaystyle G}$ ${\ displaystyle N}$

If a subgroup fulfills one of the properties mentioned above, then the subgroup is called normal or a normal subgroup, the terms normal subgroup and normal subgroup are synonymous. The notation means “ is the normal divisor of ”. Some authors also use this and reserve the name in case . ${\ displaystyle N}$${\ displaystyle N \ vartriangleleft G}$${\ displaystyle N}$${\ displaystyle G}$${\ displaystyle N \ trianglelefteq G}$${\ displaystyle N \ vartriangleleft G}$${\ displaystyle N \ not = G}$

Examples

• Every subgroup of an Abelian group is the normal subgroup of the group and many statements about normal subgroups are trivial for Abelian groups.
• Every group has the so-called trivial normal divisors, namely the full group itself and the one-subgroup consisting only of the neutral element . All other normal factors are called non-trivial. There are groups that have no non-trivial normal divisors, these are called simple . Examples are the cyclic groups with a prime number or, as the smallest non-commutative example, the alternating group A 5 . See “ Finite Simple Group ” for more examples.${\ displaystyle \ mathbb {Z} _ {p}}$ ${\ displaystyle p}$
• In the symmetrical group S 3 , the three-element subgroup is a normal divisor. The three two-element subgroups are not normal divisors.${\ displaystyle = \ left \ {e, d, d ^ {2}, s_ {1}, s_ {2}, s_ {3} \ right \}}$${\ displaystyle N = \ {e, d, d ^ {2} \}}$${\ displaystyle \ {e, s_ {i} \}}$

Remarks

The normal part relation is not transitive , that is, from and generally does not follow . An example of this fact is the alternating group A 4 , which has a normal divisor that is isomorphic to the Klein group of four . Every two-element subgroup contained therein is a normal subgroup in , but not in . ${\ displaystyle A \ vartriangleleft B}$${\ displaystyle B \ vartriangleleft C}$${\ displaystyle A \ vartriangleleft C}$ ${\ displaystyle V}$${\ displaystyle V}$${\ displaystyle A_ {4}}$

A subgroup is a normal subgroup in if and only if its normalizer is whole . A subgroup is always a normal subgroup in its normalizer. ${\ displaystyle G}$${\ displaystyle G}$

All characteristic subgroups of a group are normal parts of the group because the conjugation of group elements is an automorphism . The converse is generally not true; for example, the two-element subgroups of the small group of four are normal but not characteristic.

Archetypes of a normal divider under a group homomorphism are again normal divisors. Images of normal dividers are generally not normal, as shown in the inclusion picture of a subgroup that is not normal dividers. The images of a normal divisor under surjective group homomorphisms are normal divisors again.

A subgroup of index 2 is always a normal subgroup. More generally, if it is a subgroup and if the index of is equal to the smallest prime number which divides the order of , then is a normal divisor. ${\ displaystyle U}$${\ displaystyle U}$${\ displaystyle G}$${\ displaystyle U}$

Normal divisors, group homomorphisms and factor groups

Factor group

The secondary classes of a normal divider form a group with the complex product , which is called the factor group from to . ${\ displaystyle N}$ ${\ displaystyle G / N}$${\ displaystyle G}$${\ displaystyle N}$

The factor group therefore consists of the secondary classes of , that is , and the product of two secondary classes is defined as a complex product . For a normal divisor of and any elements of , the complex product of two secondary classes is again a secondary class, namely . : This follows from the equality of right and left cosets (s o..) . ${\ displaystyle N}$${\ displaystyle G / N = \ {g \ cdot N \ mid g \ in G \}}$${\ displaystyle (gN) \ cdot (hN) = \ {x \ cdot y \ mid x \ in gN, y \ in hN \}}$${\ displaystyle N}$${\ displaystyle G}$${\ displaystyle g, \, h}$${\ displaystyle G}$${\ displaystyle (gN) \ cdot (hN) = (gh) N}$${\ displaystyle gN \ cdot hN = g (Nh) N = g (hN) N = (gh) (NN) = (gh) N}$

For a subgroup that is not a normal subclass, the complex product of two left (or right) minor classes is generally not a left or right minor class.

Canonical homomorphism

If there is a normal divisor, then the figure is ${\ displaystyle N \ trianglelefteq G}$

${\ displaystyle \ pi \ colon G \ to G / N, \ quad g \ mapsto g \ cdot N}$,

which maps each group element to the secondary class , a group homomorphism from into the factor group . is surjective and the core is straight . One calls this group homomorphism the canonical homomorphism . ${\ displaystyle g \ in G}$${\ displaystyle gN}$${\ displaystyle G}$${\ displaystyle G / N}$${\ displaystyle \ pi}$${\ displaystyle N}$ ${\ displaystyle G \ to G / N}$

Cores as normal divisors

The core of any group homomorphism is always a normal divisor of the group shown. To clarify the definitions, the proof is set out here. Be ${\ displaystyle \ operatorname {ker} (\ varphi)}$${\ displaystyle \ varphi}$

 ${\ displaystyle \ varphi \ colon G \ to H}$ a group homomorphism and ${\ displaystyle \ operatorname {ker} (\ varphi): = \ {n \ in G \ mid \ varphi (n) = e_ {H} \}}$ its core (with as the neutral element of ). ${\ displaystyle e_ {H}}$${\ displaystyle H}$

Then is for everyone and${\ displaystyle g \ in G}$${\ displaystyle n \ in \ operatorname {ker} (\ varphi)}$

${\ displaystyle \ varphi (g \, n \, g ^ {- 1}) = \ varphi (g) \; \ varphi (n) \; \ varphi (g ^ {- 1}) = \ varphi (g) \, e_ {H} \, \ varphi (g ^ {- 1}) = \ varphi (g) \, \ varphi (g ^ {- 1}) = \ varphi (g \; g ^ {- 1}) = \ varphi (e_ {G}) = e_ {H},}$

thus and thus a normal divisor in according to definition 2. ${\ displaystyle g \, n \, g ^ {- 1} \ in \ operatorname {ker} (\ varphi)}$${\ displaystyle \ operatorname {ker} (\ varphi)}$${\ displaystyle G}$

Together with the considerations on canonical homomorphism, these considerations show that the normal divisors are precisely the kernels of group homomorphisms. In a group, congruence relations correspond exactly to the normal divisors. On this topic see also Homomorphism Theorem ”.

Normal divider and subgroup association

The normal divisors of a group form a system of sets that is even a shell system . This envelope system is a complete bandage , the normal divider bandage . Here this means specifically: ${\ displaystyle G}$

1. The intersection of normal parts of is a normal part,${\ displaystyle G}$
2. For every subset of, there is a clearly determined smallest normal divisor that contains this set. (This operation here is the envelope operation). Special cases: The trivial normal subgroup , only the neutral element contains the group , itself is a normal subgroup. From this follows the completeness of the association.${\ displaystyle T}$${\ displaystyle G}$${\ displaystyle {\ mathcal {N}} (T)}$${\ displaystyle {\ mathcal {N}}}$ ${\ displaystyle \ {e \}}$${\ displaystyle e}$${\ displaystyle {\ mathcal {N}} (\ emptyset)}$${\ displaystyle {\ mathcal {N}} (G) = G}$

As Dedekind's modular law shows, the normal divider lattice is a modular sub-lattice of the subgroup lattice. The latter is generally not modular, see " Modular group (M group) ".

Complementary normal divisors and inner direct product

In general there are no complementary objects in the normal divider lattice. However, if a normal subdivision has a complementary object , that is, applies to the normal subdivisions and , then the group can be represented as the (inner) direct product of these normal subdivisions:, that is, each group element has a unique representation as the product of elements and . Conversely, every factor of an (external) direct product is (isomorphic to a) normal divisor of the product group and the product of the other factors is isomorphic to a normal divisor that is complementary thereto. ${\ displaystyle N_ {1}}$${\ displaystyle N_ {2}}$${\ displaystyle N_ {1} \ cap N_ {2} = \ {e \}}$${\ displaystyle {\ mathcal {N}} (N_ {1} \ cup N_ {2}) = G}$${\ displaystyle G}$${\ displaystyle G \ cong N_ {1} \ times N_ {2}}$${\ displaystyle g \ in G}$${\ displaystyle g = n_ {1} \ cdot n_ {2}}$${\ displaystyle n_ {1} \ in N_ {1}}$${\ displaystyle n_ {2} \ in N_ {2}}$${\ displaystyle H_ {j}}$${\ displaystyle G = H_ {1} \ times H_ {2} \ cdots \ times H_ {n}}$${\ displaystyle G}$

A generalization of this statement: For two normal divisors that have a trivial intersection, i. H. , applies: ${\ displaystyle N_ {1} \ cap N_ {2} = \ {e \}}$

• Their elements commute with each other without one of the two normal divisors having to be commutative:
${\ displaystyle n_ {1} \ cdot n_ {2} = n_ {2} \ cdot n_ {1} \ quad {\ text {if}} \; n_ {1} \ in N_ {1}, \, n_ { 2} \ in N_ {2}}$
• Its supremum in the union of the normal divisors agrees with its complex product, which in turn is isomorphic to its (outer) direct product:
${\ displaystyle {\ mathcal {N}} (N_ {1} \ cup N_ {2}) = N_ {1} \ cdot N_ {2} \ cong N_ {1} \ times N_ {2}}$

Both statements generally do not apply to subgroups that are not normal subgroups. For example, in the free group over two elements the two infinite cyclic subgroups intersect and in the one group. The group (external direct product) is not isomorphic to any subgroup . The complex product is not a subgroup of , since e.g. B. is, but . ${\ displaystyle F = \ langle a, b \ rangle}$${\ displaystyle A = \ langle a \ rangle}$${\ displaystyle B = \ langle b \ rangle}$${\ displaystyle A \ times B}$${\ displaystyle F}$${\ displaystyle A \ cdot B}$${\ displaystyle F}$${\ displaystyle from \ in A \ cdot B}$${\ displaystyle (ab) ^ {2} = abab \ not \ in A \ cdot B}$

Inner semi-direct product

If there is only one normal subgroup and a not necessarily normal subgroup of the group and the two intersect in the one group , then the following applies: ${\ displaystyle N}$${\ displaystyle H}$${\ displaystyle G}$${\ displaystyle N \ cap H = \ {e \}}$

• The complex product is a (not necessarily normal) subgroup of .${\ displaystyle U = N \ cdot H}$${\ displaystyle G}$
• Each element is as a product of elements and displayed clearly.${\ displaystyle u \ in U}$${\ displaystyle u = n \ cdot h}$${\ displaystyle n \ in N}$${\ displaystyle h \ in H}$
• Of course, the normal divisor of always normal is in . The subgroup is normal in if and only if the elements commute from and among each other (see above).${\ displaystyle N}$${\ displaystyle G}$${\ displaystyle U}$${\ displaystyle H ${\ displaystyle U}$${\ displaystyle N}$${\ displaystyle H}$

In the situation described ( ), the complex product is called the (inner) semi-direct product of the subgroups and . The outer semidirect product consists, as stated in the article mentioned, of the Cartesian product of two groups (here and ) together with a homomorphism of into the group of automorphisms of . The outer semi-direct product is then often written as. Of the technical details, we are only interested in the fact that the calculation rule (relation) ${\ displaystyle N \ vartriangleleft G, \; H ${\ displaystyle U = N \ cdot H}$${\ displaystyle N}$${\ displaystyle H}$${\ displaystyle N}$${\ displaystyle H}$${\ displaystyle \ theta \ colon H \ to \ operatorname {Aut} (N)}$${\ displaystyle H}$${\ displaystyle N}$${\ displaystyle A = N \ rtimes _ {\ theta} H}$${\ displaystyle \ theta}$

${\ displaystyle \ left (e_ {N}, h \ right) \ cdot \ left (n, e_ {H} \ right) = \ left (\ theta (h) (n), h \ right)}$

is introduced on the Cartesian product . The notation here means that the automorphism is applied to, it always applies here as in the following . This calculation rule makes it possible to bring all products to the standard shape (by sliding the elements through from to the right) . In our case of an inner product, this corresponds to the calculation rule ${\ displaystyle N \ times H}$${\ displaystyle \ theta (h) (n)}$${\ displaystyle \ theta (h)}$${\ displaystyle n}$${\ displaystyle n \ in N, h \ in H}$${\ displaystyle H}$${\ displaystyle (n, e_ {H}) \ cdot (e_ {N}, h)}$

${\ displaystyle h \ cdot n = h \ cdot n \ cdot \ left (h ^ {- 1} \ cdot h \ right) = \ left (h \ cdot n \ cdot h ^ {- 1} \ right) \ cdot h = \ theta (h) (n) \ cdot h}$,

that is, operates on by conjugation, is the automorphism of the normal subdivision defined by this conjugation . In the sense of these considerations, the complex product (here an inner semidirect product) is isomorphic to the outer semidirect product . ${\ displaystyle H}$${\ displaystyle N}$${\ displaystyle \ theta (h) \ in \ operatorname {Aut} (N)}$${\ displaystyle N}$${\ displaystyle U}$${\ displaystyle A = N \ rtimes _ {\ theta} H}$

Every direct product is also a special semidirect product, as described here is the (inner) direct product of and , if and only if one of the following, pairwise equivalent, conditions applies: ${\ displaystyle U}$${\ displaystyle N}$${\ displaystyle H}$

• ${\ displaystyle H \ vartriangleleft U}$(also is a normal part of the product).${\ displaystyle H}$
• ${\ displaystyle \ forall n \ in N \, \ forall h \ in H \ colon \; nh = hn}$ (Elements of the two factor groups can be interchanged in products without changing the value of the product).
• ${\ displaystyle \ forall h \ in H \ colon \; \ theta (h) = \ operatorname {Id} _ {N}}$(Conjugation with elements from leaves pointwise fixed).${\ displaystyle H}$${\ displaystyle N}$