Paper 3, Section II, E

Let $p$ be a prime number, and $a$ an integer with $1 \leqslant a \leqslant p-1$. Let $G$ be the Cartesian product

$G=\{(x, u) \mid x \in\{0,1, \ldots, p-2\}, u \in\{0,1, \ldots, p-1\}\}$

Show that the binary operation

$(x, u) *(y, v)=(z, w)$

where

$\begin{aligned} z & \equiv x+y(\bmod p-1) \\ w & \equiv a^{y} u+v(\bmod p) \end{aligned}$

makes $G$ into a group. Show that $G$ is abelian if and only if $a=1$.

Let $H$ and $K$ be the subsets

$H=\{(x, 0) \mid x \in\{0,1, \ldots, p-2\}\}, \quad K=\{(0, u) \mid u \in\{0,1, \ldots, p-1\}\}$

of $G$. Show that $K$ is a normal subgroup of $G$, and that $H$ is a subgroup which is normal if and only if $a=1$.

Find a homomorphism from $G$ to another group whose kernel is $K$.

*Typos? Please submit corrections to this page on GitHub.*