Let (G, .) be a finite group (written multiplicatively), say with elements Xl' x 2' , xn ,

and let R be an arbitrary ring. Consider the set R(G) of all formal sums

(ri E R).

Two such expressions are regarded as equal if they have the same coefficients.

Addition and multiplication can be defined in R(G) by taking

n n n

I rixi + I SiXi = I (ri + S;)Xi i=l i=l i=l

and

(.f rixi) (.f SiXi) = .f tixi, 1=1 1=1 ,=1

where

ti = I rjSk' XjXk=Xi

(The meaning of the last-written sum is that the summation is to be extended over

all subscripts j and k for which XjXk = Xi') Prove that, with respect to these

operations, R(G) constitutes a ring, the so-called group ring of Gover R.