On Projective representations of direct product of groups

Abstract

Let $G=G_1\times G_2$ be a finite group. We know that the second cohomology group $H^2(G,{\mathbb{C}}^{\times })$ is isomorphic to $H^2(G_1,{\mathbb{C}}^{\times })\times H^2(G_2,{\mathbb{C}}^{\times })\times Hom(G_1/G_1^{\prime }{\otimes }_{\mathbb{Z}}{G}_2/G_2^{\prime },{\mathbb{C}}^{\times }).$ A $2$-cocycle $\alpha$ of $G$ is called a bilinear cocycle if the corresponding cohomology class $[\alpha ]$ of $H^2(G,{\mathbb{C}}^{\times })$ lies in $Hom(G_1/G_1^{\prime }{\otimes }_{\mathbb{Z}}{G}_2/G_2^{\prime },{\mathbb{C}}^{\times })$. In this article, our aim is to construct an irreducible complex projective representation $\rho$ of $G$ for bilinear cocycles $\alpha$. If $G_1$ is any abelian $p$-group and $G_2$ is an elementary abelian $p$-group, then we give a construction of $\rho$ for bilinear cocycles $\alpha$ of $G$. For a subgroup $H$ of $G$ of index $\le p^2$, we also count the number of cohomology classes $[\alpha ]$ for which the irreducible projective representations behave the same while restricting on $H$. Finally, we consider any $p$-group $G=G_1\times G_2$, and we discuss how the above construction helps us to describe an irreducible $\alpha$-representation of $G$ when $[\alpha ]$ is of order $p$ or $G_2/G_2^{\prime }$ is elementary abelian. We also discuss several examples as an application of the above results.

16 pages