Normal subgroup

A normal subgroup (also an invariant subgroup or a normal subgroup ) is a subgroup of a special type whose left and right adjacent classes coincide. Such groups are important because they allow you to build a factor group .

Content

1 Definitions
2 Examples
3 Properties
4 Historical facts
5 Links

Definitions

Subgroup $N$ ${\ displaystyle N}$ groups $G$ ${\ displaystyle G}$ is called normal if it is invariant under conjugation, that is, for any element $n$ ${\ displaystyle n}$ of $N$ ${\ displaystyle N}$ and any $g$ ${\ displaystyle g}$ of $G$ ${\ displaystyle G}$ element $gng^{-1}$ ${\ displaystyle gng ^ {- 1}}$ lies in $N$ ${\ displaystyle N}$ :

N\triangleleft G\,\iff \,\forall \,n\in N,\forall \ g\in G

{\ displaystyle N \ triangleleft G \, \ iff \, \ forall \, n \ in N, \ forall \ g \ in G}

gng^{-1}\in {N}.

{\ displaystyle gng ^ {- 1} \ in {N}.}

The following normal conditions for a subgroup are equivalent:

For anyone $g$ ${\ displaystyle g}$ of $G$ ${\ displaystyle G}$ $gNg^{-1}\subseteq N$ ${\ displaystyle gNg ^ {- 1} \ subseteq N}$ .
For anyone $g$ ${\ displaystyle g}$ of $G$ ${\ displaystyle G}$ $gNg^{-1}=N$ ${\ displaystyle gNg ^ {- 1} = N}$ .
Lots of left and right adjacent classes $N$ ${\ displaystyle N}$ at $G$ ${\ displaystyle G}$ match.
For anyone $g$ ${\ displaystyle g}$ of $G$ ${\ displaystyle G}$ $gN=Ng$ ${\ displaystyle gN = Ng}$ .
$N$ ${\ displaystyle N}$ isomorphic to the union of conjugate classes.

Condition (1) is logically weaker than (2), and condition (3) is logically weaker than (4). Therefore, conditions (1) and (3) are often used to prove the normality of a subgroup, and conditions (2) and (4) are used to prove the consequences of normality.

Examples

$\{e\}$ ${\ displaystyle \ {e \}}$ and $G$ ${\ displaystyle G}$ - always normal subgroups $G$ ${\ displaystyle G}$ . They are called trivial. If there are no other normal subgroups, then the group $G$ ${\ displaystyle G}$ called simple .

The center of the group is a normal subgroup.

The commutant of a group is a normal subgroup.

Any characteristic subgroup is normal, since conjugation is always an automorphism .

All subgroups $N$ ${\ displaystyle N}$ abelian group $G$ ${\ displaystyle G}$ normal since $gN=Ng$ ${\ displaystyle gN = Ng}$ . A non-Abelian group in which any subgroup is normal is called Hamiltonian .

A group of parallel translations in a space of any dimension is a normal subgroup of a Euclidean group ; for example, in three-dimensional space, turning, shifting and turning in the opposite direction leads to a simple shift.

In the group of a Rubik's cube, a subgroup consisting of operations that act only on corner elements is normal, since no conjugate transformation will force such an operation to act on a boundary, rather than an corner element. On the contrary, a subgroup consisting only of rotations of the upper face is not normal, since conjugations allow moving parts of the upper face down.

Properties

Normality is preserved under surjective homomorphisms and taking inverse images.
The kernel of a homomorphism is a normal subgroup.
Normality is preserved when constructing a direct product .
A normal subgroup of a normal subgroup does not have to be normal in a group, that is, normality is not transitive . However, the characteristic subgroup of a normal subgroup is normal.
Each subgroup of index 2 is normal. If $p$ ${\ displaystyle p}$ - smallest prime order divider $G$ ${\ displaystyle G}$ , then any subgroup of the index $p$ ${\ displaystyle p}$ is normal.
If $N$ ${\ displaystyle N}$ Is a normal subgroup in $G$ ${\ displaystyle G}$ , then on the set of left (right) adjacent classes $G/N$ ${\ displaystyle G / N}$ you can enter a group structure according to the rule

(g_{1}N)(g_{2}N)=(g_{1}g_{2})N

{\ displaystyle (g_ {1} N) (g_ {2} N) = (g_ {1} g_ {2}) N}

The resulting set is called the quotient group.

G

{\ displaystyle G}

by

N

{\ displaystyle N}

.

$N$ ${\ displaystyle N}$ is normal if and only if it acts trivially on left adjoining classes $G/N$ ${\ displaystyle G / N}$ .
Each normal subgroup is quasinormal.

Historical Facts

Evarist Galois was the first to understand the importance of normal subgroups.

Links

Vinberg E. B. Algebra Course - M .: Factorial Press Publishing House, 2002, ISBN 5-88688-060-7
Kostrikin A.I. Introduction to Algebra. Part III. The main structure. - 3rd ed. - M .: FIZMATLIT, 2004 .-- 272 p. - ISBN 5-9221-0489-6 .