Past exam of the groups and representations course of the University of Oxford / 2019 / 1 / b

Nonzero scalar matrices form a subgroup $N$ of the general linear group, since products and inverses remain nonzero scalar matrices. Moreover, for every invertible $g$,Thus $N$ lies in the center of a group and is a normal subgroup. The quotient group is the complex projective linear group ${\mathrm{PGL}}_n(\mathbb{C})$, whose elements identify invertible matrices differing by a nonzero scalar.

The canonical group homomorphism $\pi :g\mapsto [g]$ obeys $[g][h]=[gh]$. Composing it with the given group representation givesConsequentlyThis is the projective representation associated with $R$: a nonzero scalar acts trivially on each line in the projective space of $V$. Its kernel of a group homomorphism consists of those $g$ for which $R(g)$ is scalar, which can be larger than the kernel of a group homomorphism of $R$ itself.