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
- Normal automorphism
- Normality can be written in terms of weakly normal automorphisms:
- Weakly normal automorphism
Function
- Weakly normal automorphism
Endomorphism
- Weakly normal automorphism
Injective endomorphism
- Weakly normal automorphism
- Normality can be written in terms of monomial automorphisms:
- Monomial automorphism
Function
- Monomial automorphism
Endomorphism
- Monomial automorphism
Injective endomorphism
- Monomial automorphism
- Normality can be written in terms of strong monomial automorphisms:
- Strong monomial automorphism
Function
- Strong monomial automorphism
Endomorphism
- Strong monomial automorphism
Automorphism
- Strong monomial 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