Group ring

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 𝐺 be a group, 𝑅 be a ring, and 𝑅[𝐺] denote the Group ring of maps 𝐺 →𝑅 of finite support. Then 𝑓 ∈𝑍(𝑅[𝐺]) (Centre of a rng) iff 𝑓 is a Group class function, group i.e. 𝑓 ∗𝑔 =𝑔 ∗𝑓 for all 𝑔 ∈𝑅[𝐺] iff 𝑓(𝑦𝑥𝑦−1) =𝑓(𝑥) for all 𝑥,𝑦 ∈𝐺.

Proof

Let 𝑓 ∗𝑔 =𝑔 ∗𝑓 for all 𝑔 ∈𝑅[𝐺] Since {𝛿𝑧}𝑧∈𝐺 ={𝛿𝑧−1}𝑧∈𝐺 forms a basis of the group ring, any 𝑔 may be expressed as

𝑔=∑𝑧∈𝐺𝑔(𝑧−1)𝛿𝑧−1

and thus for all 𝑔 ∈𝑅[𝐺] and 𝑥 ∈𝐺

𝑓∗(∑𝑧∈𝐺𝑔(𝑧−1)𝛿𝑧−1)=(∑𝑧∈𝐺𝑔(𝑧−1)𝛿𝑧−1)∗𝑓∑𝑤∈𝐺∑𝑧∈𝐺𝑓(𝑥𝑤−1)𝑔(𝑧−1)𝛿𝑧−1(𝑤)=∑𝑤∈𝐺∑𝑧∈𝐺𝑔(𝑧−1)𝛿𝑧−1(𝑥𝑤−1)𝑓(𝑤)∑𝑧∈𝐺𝑓(𝑥𝑧)𝑔(𝑧−1)=∑𝑧∈𝐺𝑓(𝑧𝑥)𝑔(𝑧−1)

which is true iff 𝑓(𝑥𝑧) =𝑓(𝑧𝑥) for all 𝑥,𝑧 ∈𝐺, which in turn is true iff 𝑓(𝑧𝑥𝑧−1) =𝑓(𝑥) for all 𝑥,𝑧 ∈𝐺.

Thus dim⁡𝑍(𝑅[𝐺]) equals the number of conjugacy classes.

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