Projective representations of a group $G$ are classified by the second group cohomology $H^2(G,U(1))$. If $G$ is finite and abelian, it is isomorphic to the direct product of cyclic groups


$$
G\cong C\_{n\_1,...,n\_k}:=\mathbb{Z}\_{n\_1}\times ...\times \mathbb{Z}\_{n\_k}
$$


Hence I was wandering if there is any known result for


$$
H^2(C\_{n\_1,...,n\_k},U(1))
$$


I know that
$$
H^2(\mathbb{Z}\_n,U(1))=0 \ , \ \ \ \ \ \ H^2(\mathbb{Z}\_{n\_1}\times \mathbb{Z}\_{n\_2},U(1))\cong \mathbb{Z}\_{\text{gcd}(n\_1,n\_2)}
$$
but I don't know about are a general result or even a result for $k=3$.



Projective representations of a finite abelian group
The answer is $\ \displaystyle\bigoplus\_{i<j}\ \mathbb{Z}/\!\gcd(n\_i,n\_j)$. The reason is that for $G$ finite, $H^2(G,U(1))$ is the dual abelian group of $H\_2(G,\mathbb{Z})$. Now use the Künneth formula.


