Centre of the group ring

The following theorem means we can speak of class functions into some ring as the centre of the group ring:

Let be a group, be a ring, and denote the Group ring of maps of finite support. Then (Centre of a rng) iff is a Group class function, #m/thm/group i.e. for all iff for all .

Proof

Let for all Since forms a basis of the group ring, any may be expressed as

and thus for all and

which is true iff for all , which in turn is true iff for all .

Thus equals the number of conjugacy classes.

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