Group theory MOC

Centre of a group

The centre of a group is a Normal subgroup of all elements of that commute with every other element, i.e. . Note that at the very least .¹ group group

Proof of normal subgroup

As shown above, . Additionally, for any it is clear that since , so is closed under the binary operation. Since for any we can both pre- and postmultiply both sides to obtain for any , therefore , so is closed under the inverse. Hence is a subgroup of by Two step subgroup test. Now let and . Clearly . Hence is a Normal subgroup.

A related notion is the Centralizer in a group. The centre is the intersection of all centralisers.

Properties

• The centre is necessarily abelian.

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Footnotes

1. 2017, Contemporary Abstract Algebra, pp. 67 ↩