Centraliser: Difference between revisions
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In [[group theory]], the '''centraliser''' of a [[subset]] of a [[group (mathematics)]] is the set of all group elements which [[commutativity|commute]] with every element of the given subset. | In [[group theory]], the '''centraliser''' of a [[subset]] of a [[group (mathematics)]] is the set of all group elements which [[commutativity|commute]] with every element of the given subset. | ||
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:<math> C_G(S) = \{ g \in G : \forall s \in S,~ gs=sg \} . \, </math> | :<math> C_G(S) = \{ g \in G : \forall s \in S,~ gs=sg \} . \, </math> | ||
The centraliser of any set is a [[subgroup]] of ''G'', and the centraliser of ''S'' is equal to the centraliser | The centraliser of any set is a [[subgroup]] of ''G'', and the centraliser of ''S'' is equal to the centraliser of the subgroup <math>\langle S \rangle</math> generated by the subset ''S''. | ||
The centraliser of the [[empty set]] is the whole group ''G''; the centraliser of the whole group ''G'' is the [[centre of a group|centre]] of ''G''. | The centraliser of the [[empty set]] is the whole group ''G''; the centraliser of the whole group ''G'' is the [[centre of a group|centre]] of ''G''.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 11:00, 26 July 2024
In group theory, the centraliser of a subset of a group (mathematics) is the set of all group elements which commute with every element of the given subset.
Formally, for S a subset of a group G, we define
The centraliser of any set is a subgroup of G, and the centraliser of S is equal to the centraliser of the subgroup generated by the subset S.
The centraliser of the empty set is the whole group G; the centraliser of the whole group G is the centre of G.