Normaliser: Difference between revisions
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In [[group theory]], the '''normaliser''' of a [[subgroup]] of a [[group (mathematics)]] is the set of all group elements which map the given subgroup to itself by [[Conjugation (group theory)|conjugation]]. | In [[group theory]], the '''normaliser''' of a [[subgroup]] of a [[group (mathematics)]] is the set of all group elements which map the given subgroup to itself by [[Conjugation (group theory)|conjugation]]. | ||
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A subgroup of ''G'' is [[normal subgroup|normal]] in ''G'' if its normaliser is the whole of ''G''. | A subgroup of ''G'' is [[normal subgroup|normal]] in ''G'' if its normaliser is the whole of ''G''. | ||
The normaliser of the [[trivial subgroup]] is the whole group ''G''. | The normaliser of the [[trivial subgroup]] is the whole group ''G''.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:00, 26 September 2024
In group theory, the normaliser of a subgroup of a group (mathematics) is the set of all group elements which map the given subgroup to itself by conjugation.
Formally, for H a subgroup of a group G, we define
A subgroup of G is normal in G if its normaliser is the whole of G.
The normaliser of the trivial subgroup is the whole group G.