# Difference between revisions of "Normal subgroup"

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

Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. | Term variations: [[Groupprops:Subnormal subgroup|subnormal subgroup]], [[Groupprops:Abnormal subgroup|abnormal subgroup]], [[Groupprops:Quasinormal subgroup|quasinormal subgroup]], and others. See [[Groupprops:Category:Variations of normality]], [[Groupprops:Category:Opposites of normality]], and [[Groupprops:Category:Analogues of normality]]. |

## Revision as of 01:16, 21 March 2009

**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.

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