Given a group $G$ and a ring $R$, the underlying set of $R[G]$ is the set of finite linear combinations using $G$ as a basis. Addition: $(\sum_g a_gg)(\sum_h b_hh)=\sum_k (a_k+b_k)k$. Multiplication: $(\sum_g a_gg)(\sum_h b_hh):=\sum_{gh=k}a_gb_hk$
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