$$\C{\S}=\bigoplus_{ s\in {\S}}\C e(s)$$ which is an associative algebra under product $e(s)e(t)=\d_{s,t}e(t)$ and isomorphic to $\C^{|S|}.$ The right action of $G$ on $\S$ induces a right $G$-module structure on $\C{\S}.$ Let $${\U}(\C{\S})=\{\sum_{ s\in {\S} }\l_{s} e(s)|\l_{s} \in \C^{*}\}$$ be the set of units of $\C{\S}.$ It is easy to check that $\U(\C{\S})$ is a multiplicative right $G$-module.

From now on we will fix an element $\a\in Z^2(G,\U(\C\S))$ such that $\a(h,1)=\a(1,h) =\sum_{s\in {\S}}e(s)$ for all $h$ in $G$. Then $\a$ defines functions $\a_s: G\times G\to \C^*$ for $s\in {\S}$ such that $$\a(h,k)=\sum_{s\in S}\a_s(h,k)e(s)$$ for $h,k\in G.$ The cocycle condition $\a(hk,l)\a(h,k)^l=\a(h,kl)\a(k,l)$ implies immediately that $$\a_s(hk,l)\a_{(s)l^{-1}}(h,k)=\a_s(h,kl)\a_s(k,l)$$ for $h,k,l\in G$ and $s\in\S.$

The main object that we study in this section is the vector space ${\A}_{\a}(G,{\S})=\C[G]\otimes \C\S$ with a basis $g\o e(s)$ for $g\in G$, and $s\in {\S}$.

i\) The ${\A}_{\a}(G,{\S})$ is an associative algebra under multiplication defined by $$g\o e(s)\cdot h\o e(t)=\a_t(g,h)gh\o e(sh)e(t).$$

ii\) If $\a$ is cohomologous to $\b$ then ${\A}_{\a}(G,{\S})\cong{\A}_{\b}(G,{\S})$ as algebras.

For i), we first prove that the product is associative. Let $g,h,k\in G$ and $s,t,u\in \S.$ Then $$\begin{aligned}
& &\ \ \ \ (g\o e(s)\cdot h\o e(t))\cdot k\o e(u)\\
& &=(\a_t(g,h)gh\o e(sh)e(t))\cdot k\o e(u)\\
& &=\d_{sh,t}\a_t(g,h)gh\o e(t)\cdot k\o e(u)\\
