Past exam of the mathematics course of the University of Cambridge / 2011 / ia / Paper 3 / 6D / b / Solution

The general linear group $G{L}_2({\mathbb{F}}_5)$ consists of invertible two-by-two matrices over the finite field ${\mathbb{F}}_5$, under matrix multiplication. Such a matrix has two linearly independent columns. There are $5^2-1=24$ choices for a nonzero first column, and $5^2-5=20$ choices for a second column outside its one-dimensional span. ThusThe determinant is a group homomorphismwhere ${\mathbb{F}}_5^{\times }=\{1,2,3,4\}$ is the multiplicative group of nonzero field elements. It is surjective because $\mathrm{diag}(t,1)$ has determinant $t$. Its kernel of a group homomorphism is $S{L}_2({\mathbb{F}}_5)$. The first isomorphism theorem gives a quotient of order four, whenceThe nonzero squares form the subgroup $\{1,4\}$ of ${\mathbb{F}}_5^{\times }$. Its inverse imageis a subgroup with index two. Indeed, composing the determinant with the quotient map to ${\mathbb{F}}_5^{\times }/\{1,4\}$ makes $H$ the kernel of a surjective group homomorphism to a group of order two. Thus $|H|=240$.