The notation of ring $R (G)$ in Serre's Linear Representations of Finite Groups

Question

I am trying to understand the expression for $R (G)$ in the first paragraph (Section 9.1) of Chapter 9 in Serre's Linear Representations of Finite Groups.

Let me reproduce it here.

$$ R (G) = \mathbb{Z} \chi_1 \oplus \ldots \oplus \mathbb{Z} \chi_h $$.

I understand that in the paragraph $R(G)$ is referred to as a group. I am not familiar with this particular presentation of group. Could anyone please help me out on how to understand the group elements and operation from this presentation?

Thanks in advance!

Answer 1

You do know that for a finite group $G$, the set of class functions on $G$ is an abelian group under addition.

Then in Section 9.1 of Serre's Linear Representations of Finite Groups, the group $R(G)$ is defined as the subgroup generated by the irreducible characters of $G$ . Then it is a consequence of orthonormality that $R(G)$ is a free abelian group with basis given by the set $\{\chi_1, \ldots, \chi_h\}$ irreducible characters. That is, $$R(G) = \mathbb{Z} \chi_1 \oplus \cdots \oplus \mathbb{Z} \chi_h.$$

I am not sure what is your main confusion here. If it is just that the notation is unfamiliar, look up "direct sum", "finitely generated abelian group", "free abelian group", etc. in google.