On Projective representations of direct product of groups

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 $H^2(G_1,\mathbb C^\times) \times H^2(G_2,\mathbb C^\times) \times Hom(G_1/G_1' \otimes_\mathbb Z G_2/G_2', \mathbb C^\times ).$ 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 $Hom(G_1/G_1' \otimes_\mathbb Z G_2/G_2', \mathbb C^\times)$. 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 $\leq p^2$, 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 $G_2/G_2'$ is elementary abelian. We also discuss several examples as an application of the above results.Comment: 16 page

On Projective Representations of Finitely Generated Groups

For a finitely generated group G, we construct a representation group of G explicitly. Along the way, we describe the structure of the 2-cocycles of G. We also prove a characterization of monomial projective representations of finitely generated nilpotent groups and a characterization of polycyclic groups whose projective representations are finite dimensional.Comment: 22 page

Distorted wurtzite unit cells: Determination of lattice parameters of non-polar a-plane AlGaN and estimation of solid phase Al content

Unlike c-plane nitrides, ``non-polar" nitrides grown in e.g. the a-plane or m-plane orientation encounter anisotropic in-plane strain due to the anisotropy in the lattice and thermal mismatch with the substrate or buffer layer. Such anisotropic strain results in a distortion of the wurtzite unit cell and creates difficulty in accurate determination of lattice parameters and solid phase group-III content (x_solid) in ternary alloys. In this paper we show that the lattice distortion is orthorhombic, and outline a relatively simple procedure for measurement of lattice parameters of non-polar group III-nitrides epilayers from high resolution x-ray diffraction measurements. We derive an approximate expression for x_solid taking into account the anisotropic strain. We illustrate this using data for a-plane AlGaN, where we measure the lattice parameters and estimate the solid phase Al content, and also show that this method is applicable for m-plane structures as well