Funky definitions of normal subgroup

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This is a survey article related to:normal subgroup
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1 Definition in terms of measures of deviation from normality
2 Definition in terms of group actions
3 Definitions from category theory
4 Definitions from universal algebra
5 Definition in terms of linear representations
6 Definitions in terms of generating sets
7 Definition via function restriction expression
- 7.1 Extensible automorphism
8 Circular definitions

Definition in terms of measures of deviation from normality

A normal subgroup is a subgroup whose normalizer is the whole group.
A normal subgroup is a subgroup that equals its normal core.
A normal subgroup is a subgroup that equals its normal closure.
A normal subgroup is a subgroup that equals each of its conjugate subgroups.

Definition in terms of group actions

A normal subgroup is a fixed point under the action of the group on the set of its subgroups by conjugation.

Definitions from category theory

A normal subgroup is a normal monomorphism (i.e., it is a monomorphism that occurs as the kernel of some epimorphism). Here, we are thinking of subgroups not as subsets but rather as inclusion maps.

Definitions from universal algebra

A normal subgroup is a kernel in the variety with zero of groups.
A normal subgroup is an ideal in the variety with zero of groups.

Definition in terms of linear representations

A subgroup $N$ of a group $G$ is termed normal in $G$ if and only if there exists a linear representation of $G$ over a field of characteristic zero, with the property that the character of the representation is nonzero on all elements of $N$ , and zero on all elements outside $N$ .
A subgroup $N$ of a group $G$ is termed normal in $G$ if and only if the trivial linear representation of $N$ over characteristic zero, induces a representation of $G$ (by induction of representations) that is zero on all elements outside $N$ .

Definitions in terms of generating sets

A subgroup $N$ of a group $G$ is normal in $G$ if, whenever $A$ is a generating set of $G$ and $B$ is a generating set of $N$ , $aba^{-1} \in N$ and $a^{-1}ba \in N$ for all $a \in A, b \in B$ . (Note that this is the definition used to test normality -- Further information: Normality testing problem
Suppose $G$ is a group and $N$ is a subgroup. Suppose we quotient out $G$ by relations of the form $g = e$ for all $g \in N$ . Then $N$ is normal if and only if the only elements that become trivial in the quotient, are those that originally came from $N$ .

Definition via function restriction expression

Extensible automorphism

Further information: Extensible automorphism, extensible implies subgroup-conjugating, extensible implies normal, extensible automorphism-invariant equals normal

Here are some quick definitions:

An extensible automorphism of a group is an automorphism that can be extended to an automorphism of any bigger group containing it.
An infinity-extensible automorphism of a group is an automorphism that can be extended to an infinity-extensible automorphism of any bigger group containing it (in other words, it can be extended infinitely often).
A pushforwardable automorphism of a group is an automorphism that can be pushed forward via any homomorphism.

Every infinity-extensible automorphism of a group is extensible and every extensible automorphism of a group is subgroup-conjugating: it sends every subgroup to a conjugate subgroup. In particular, every extensible automorphism of a group is normal: it preserves each normal subgroup. Thus, normality has the following function restriction expressions:

As the invariance property with respect to extensible automorphisms:

Extensible automorphism $\to$ Function

In other words, a subgroup $H$ of a group $G$ is normal in $G$ if and only if every extensible automorphsim of $G$ sends $H$ to itself.

As the invariance property with respect to infinity-extensible automorphisms:

Infinity-extensible automorphism $\to$ Function

In other words, a subgroup $H$ of a group $G$ is normal in $G$ if and only if every infinity-extensible automorphism of $G$ sends $H$ to itself.

As the invariance property with respect to pushforwardable automorphisms:

Pushforwardable automorphism $\to$ Function

In other words, a subgroup $H$ of a group $G$ is normal in $G$ if and only if every pushforwardable automorphism of $G$ sends $H$ to itself.

Circular definitions

A normal subgroup is a subgroup such that every characteristic subgroup of it is normal in the whole group. Further information: characteristic of normal implies normal
A normal subgroup is a subgroup that is normal in every intermediate subgroup. Further information: Normality satisfies intermediate subgroup condition
A normal subgroup is a subgroup whose intersection with every normal subgroup is normal. Further information: Normality is strongly intersection-closed
A subgroup $H$ is normal in $G$ if there exists a group $K$ containing $G$ , such that $H$ is normal in $G$ . Further information: Normality satisfies intermediate subgroup condition