# Construction of symmetry group algebra

In Kitaev's reasoning of constructing the algebra of symmetry group, he said,

"considering the symmetry group G of a fermionic system and a map $$\alpha$$ $$\alpha: G \rightarrow \mathbb{Z}_2$$ where $$\alpha$$ is the indicator that determines whether the group element is unitary or anti-unitary ,then we consider a element in the center of the $$G$$ $$p \in Z(G)$$ where $$P$$ is the fermionic parity operator $$P = (-1)^{N}$$, N is the number of particles, we can easily know that $$P$$ is unitary so that $$\alpha(P) = 1$$ ,then we can construct the "group algebra"(not sure whether it is called like this?) $$\mathscr{A} = \mathbb{R}[G]/(P+1)$$ "

First of all, what is $$\mathbb{R}[G]$$?

It seems like a clifford algebra construction but I have no idea of how these things are organized, especially how to calculate $$\mathscr{A}$$? Hope that someone could give me a logical picture of how things go.