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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 7.1 Extensible automorphism 7.2 Normal automorphisms, monomial automorphisms, strong monomial automorphisms, and weakly normal automorphisms 8 Circular definitions

Definition in terms of measures of deviation from normality

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

Definition in terms of group actions

1. 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

1. 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

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

Definition in terms of linear representations

1. 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.
2. 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

1. 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, and for all . (Note that this is the definition used to test normality -- Further information: Normality testing problem
2. Suppose G is a group and N is a subgroup. Suppose we quotient out G by relations of the form g = e for all . 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 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 H of a group G is normal in G if and only if every extensible automorphism of G sends H 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

1. 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
2. A normal subgroup is a subgroup that is normal in every intermediate subgroup. Further information: Normality satisfies intermediate subgroup condition
3. A normal subgroup is a subgroup whose intersection with every normal subgroup is normal. Further information: Normality is strongly intersection-closed
4. 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