Nonstandard definitions of normal subgroup

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 $$N$$, $$aba^{-1} \in N$$ and $$a^{-1}ba \in N$$ for all $$a \in A, b \in B$$. (Note that this is the definition used to test normality --
 * 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 $$g \in N$$. Then $$N$$ is normal if and only if the only elements that become trivial in the quotient, are those that originally came from $$N$$.

Extensible automorphism
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 $$\to$$ 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 $$\to$$ Function
 * Normal automorphism $$\to$$ Endomorphism
 * Normal automorphism $$\to$$ Automorphism


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


 * Weakly normal automorphism $$\to$$ Function
 * Weakly normal automorphism $$\to$$ Endomorphism
 * Weakly normal automorphism $$\to$$ Injective endomorphism


 * Normality can be written in terms of monomial automorphisms:


 * Monomial automorphism $$\to$$ Function
 * Monomial automorphism $$\to$$ Endomorphism
 * Monomial automorphism $$\to$$ Injective endomorphism


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


 * Strong monomial automorphism $$\to$$ Function
 * Strong monomial automorphism $$\to$$ Endomorphism
 * Strong monomial automorphism $$\to$$ Automorphism

Circular definitions

 * 1) A normal subgroup is a subgroup such that every characteristic subgroup of it is normal in the whole group.
 * 2) A normal subgroup is a subgroup that is normal in every intermediate subgroup.
 * 3) A normal subgroup is a subgroup whose intersection with every normal subgroup is normal.
 * 4) A subgroup $$H$$ is normal in $$G$$ if there exists a group $$K$$ containing $$G$$, such that $$H$$ is normal in $$G$$.