Normal (mathematics)

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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

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).

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.

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.

In Galois 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.

In commutative algebra

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

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

In differential geometry

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.

In normed vector spaces

In general normed vector spaces, as well as in tangent spaces to Riemannian manifolds, normalize means to scale by a factor to make the norm equal to 1.

In probability/statistics

Normal distribution: A particular kind of distribution parametrized by mean and variance, also called the Gaussian distribution.

Normal random variable: A random variable whose distribution is a normal distribution.