# Nonstandard definitions of normal subgroup

From Groupprops

This article lists nonstandard definitions of the following term: normal subgroup

## Contents

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

## Definitions in terms of generating sets

- A subgroup of a group is normal in if, whenever is a generating set of and is a generating set of , and for all . (Note that this is the definition used to test normality --
`Further information: Normality testing problem` - Suppose is a group and is a subgroup. Suppose we quotient out by relations of the form for all . Then is normal if and only if the only elements that become trivial in the quotient, are those that originally came from .

## Definition via function restriction expression

### Extensible automorphism

`Further information: Extensible automorphism, extensible implies inner, quotient-pullbackable implies inner`

An extensible automorphism of a group is an automorphism that can be extended to an automorphism of any bigger group containing it.

It turns out that an automorphism of a group is extensible iff it is inner. Thus, normality can be expressed as the invariance property with respect to extensible automorphisms:

Extensible automorphism Function

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

### Normal automorphisms, monomial automorphisms, strong monomial automorphisms, and weakly normal automorphisms

- Normality can be written in terms of normal automorphisms:

- Normal automorphism Function
- Normal automorphism Endomorphism
- Normal automorphism Automorphism

- Normality can be written in terms of weakly normal automorphisms:

- Weakly normal automorphism Function
- Weakly normal automorphism Endomorphism
- Weakly normal automorphism Injective endomorphism

- Normality can be written in terms of monomial automorphisms:

- Monomial automorphism Function
- Monomial automorphism Endomorphism
- Monomial automorphism Injective endomorphism

- Normality can be written in terms of strong monomial automorphisms:

- Strong monomial automorphism Function
- Strong monomial automorphism Endomorphism
- Strong monomial automorphism Automorphism

## 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 is normal in if there exists a group containing , such that is normal in .
`Further information: Normality satisfies intermediate subgroup condition`