Math 2.0

Group theory and generalizations Definition

Centre of a group

Builds on Group

Definition

Let GG be a group. The centre of GG is the set Z(G)={z∈G:zg=gz for all g∈G}Z(G) = \{z \in G : zg = gz \text{ for all } g \in G\}.

Remarks

The centre is an abelian normal subgroup of GG, and it is characteristic. The group GG is abelian if and only if Z(G)=GZ(G) = G. The centre of the symmetric group SnS_n is trivial for n≥3n \geq 3. The centre of GLn(ℝ)\mathrm{GL}_n(\mathbb{R}) is the set of the nonzero scalar matrices λI\lambda I.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Center_(group_theory), lead section
  2. Web https://en.wikipedia.org/wiki/Center_(group_theory), As a subgroup
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