Centraliser: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Richard Pinch
(new entry, just a stub)
 
imported>Richard Pinch
m (links)
Line 5: Line 5:
:<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 of of the subgroup <math>\langle S \rangle</math> generated by the subset ''S''.
The centraliser of any set is a [[subgroup]] of ''G'', and the centraliser of ''S'' is equal to the centraliser of 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''.

Revision as of 15:47, 5 December 2008

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 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.