# Normal (mathematics)

The term *normal* in mathematics is used in the following broad senses:

- To denote something upright or perpendicular
- To denote something that is as it
*should be*. In this sense, normal means*good*or*desirable*rather than*typical*

It is typically used as a limiting adjective, and in a binary sense. In other words, an object in a certain class is *normal* if it satisfies certain conditions, and every object in that class is either normal or not normal. The noun form is *normality*. Thus, normality can be viewed as a property in various contexts.

## Contents

### In group theory

**Normal subgroup**: A subgroup of a group that occurs as the kernel of a homomorphism, or equivalently, such that every left coset and right coset are equal.

Related terms: Normality (the property of a subgroup being normal), normal core (the largest normal subgroup contained in a given subgroup), normal closure (the smallest normal subgroup containing a given subgroup), normalizer (the largest subgroup containing a given subgroup, in which it is normal), normal automorphism (an automorphism that restricts to an automorphism on every normal subgroup).

Term variations: subnormal subgroup, abnormal subgroup, quasinormal subgroup, and others. See Groupprops:Category:Variations of normality, Groupprops:Category:Opposites of normality, and Groupprops:Category:Analogues of normality.

Primary subject wiki entry: Groupprops:Normal subgroup. See also Groupprops:Questions:Normal subgroup.

Other subject wiki entries: Diffgeom:Normal subgroup

Coverage in guided tours: Groupprops guided tour for beginners; section not yet prepared.

Also located at: Wikipedia:Normal subgroup, Planetmath:NormalSubgroup, Mathworld:NormalSubgroup, Springer Online Reference Works, Citizendium:Normal subgroup

### In topology

**Normal space**: A topological space is termed normal if all points are closed sets (the assumption), and any two disjoint closed subsets can be separated by disjoint open subsets.

In some definitions, the assumption is skipped.

Related terms: normality (the property of a topological space being normal)

Term variations: Variations of normality offers a list.

Primary subject wiki entry: Topospaces:Normal space

Also located at: Wikipedia:Normal space, Planetmath:NormalSpace, Mathworld:NormalSpace

### In linear algebra

**Normal form**: also called canonical form, is a standard form in which to put a matrix (typically, upto conjugation). Two normal forms commonly used are the Jordan normal form and the rational normal form.

**Normal matrix**: A matrix over the complex numbers, that commutes with its conjugate-transpose.

No primary subject wiki entry.

Also located at: Wikipedia:Normal matrix, Mathworld:NormalMatrix, Planetmath:NormalMatrix

### In field theory

**Normal field extension**: A field extension is termed normal if the fixed field under the automorphism group of the extension, is precisely the base field.

Primary subject wiki entry: Galois:Normal extension

Also located at: Planetmath:NormalExtension, Wikipedia:Normal extension, Mathworld:NormalExtension

### In commutative algebra

**Normal domain**: An integral domain is termed normal if it is integrally closed in its field of fractions.

Primary subject wiki entry: Commalg:Normal domain

**Normal ring**: A commutative unital ring is termed normal if it is integrally closed in its total quotient ring.

Primary subject wiki entry: Commalg:Normal ring

### In differential geometry

Normal to a curve in the plane: The line perpendicular to the tangent to the curve at a point.

**Normal bundle** to a submanifold, or immersed manifold: The quotient of the tangent bundle to the whole manifold (restricted to the submanifold), by the tangent bundle to the submanifold, or immersed manifold. When the manifolds are Riemannian, the normal bundle can be viewed as the orthogonal complement to the tangent bundle of the submanifold, in the tangent bundle to the whole manifold.

Vectors in the normal bundle are termed normal vectors.

Main subject wiki entry: Diffgeom:Normal bundle

Also located at: Wikipedia:Normal bundle, Mathworld:NormalBundle, Planetmath:NormalBundle

### In category theory

**Normal monomorphism**: A monomorphism in a preadditive category, or more generally, in a category enriched over pointed sets, that arises as the kernel of an epimorphism.

Primary subject wiki entry: Cattheory:Normal monomorphism

**Normal epimorphism**: An epimorphism in a preadditive category, or more generally a category enriched over pointed sets, that arises as the cokernel of a monomorphism.

Primary subject wiki entry: Cattheory:Normal epimorphism

**Normal category**: A preadditive category, or more generally a category enriched over pointed sets, in which every monomorphism is a normal monomorphism.

Primary subject wiki entry: Cattheory:Normal category