Let $G$ be a finite $p$-group and $(\mathbb{F},\mathcal{O},\mathbb{K})$ a sufficiently large $p$-modular system. Furthermore let $\mu\_{p^\infty}\leq \mathcal{O}^\times$ the group of roots of unity of $p$-power order.


If $\alpha$ is a 2-cocyle with values in $\mu\_{p^\infty}$ does there always exist a one-dimensional projective representation $G\to\mathcal{O}^\times$ with cocylce $\alpha$ ? In other words: Does the twisted group algebra $\mathcal{O}\_\alpha[G]$ always have a one-dimensional representation $\mathcal{O}\_\alpha[G]\to\mathcal{O}$?



Are there always 1-dimensional projective representations
Ah, dammit. The answer is no and the first group I would have looked at next is the counterexample... the extra special group $G=\langle x,y,z \mid x^p=y^p=[x,y]=z, z^p=1\rangle$ fits into an non-trivial extension $1\to C\_p \to G \to C\_p\times C\_p\to 1$ and the corresponding 2-cocycle $\alpha$ gives us the counterexample, because $\mathcal{O}\_\alpha[C\_p\times C\_p] = \mathcal{O}\langle X,Y \mid X^p=Y^p=XYX^{-1}Y^{-1}=\zeta\_p\rangle$ does not have one-dimensional representations into something of characteristic zero. If $\rho$ was such a thing, $\rho(X)$ and $\rho(Y)$ would commute so that $\rho(X)\rho(Y)\rho(X)^{-1}\rho(Y)^{-1}$ would have to be both equal to $1$ and to $\zeta\_p$.


