Math 2.0

Group theory and generalizations Definition

Centraliser

Builds on Subgroup

Definition

Let GG be a group and let SS be a subset of GG. The centraliser of SS in GG is the set CG(S)={g∈G:gs=sg for all s∈S}C_G(S) = \{g \in G : gs = sg \text{ for all } s \in S\}. For an element a∈Ga \in G, the notation CG(a)C_G(a) means CG({a})C_G(\{a\}).

Remarks

The centraliser CG(S)C_G(S) is a subgroup of GG. The centraliser of GG itself is the centre Z(G)Z(G). The element aa lies in its own centraliser CG(a)C_G(a), and so does each power of aa.

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Group, concept, no evidence levelGroupconceptSubgroup, definition, no evidence levelSubgroupdefinitionCentraliser, definition, no evidence levelCentraliserdefinition
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  1. Group concept
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Where this leads

7 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Centralizer_and_normalizer, Group and semigroup
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