Twisted cyclic pair group ($(x,u)(y,v)=(x+y,a^{y}u+v)$)

Let $p$ be prime, $a\in {\mathbb{F}}_p^{\times }$, $x,y\in \mathbb{Z}/(p-1)\mathbb{Z}$ and $u,v\in {\mathbb{F}}_p$. Fermat's little theorem makes the displayed multiplication well defined. Its identity is $(0,0)$ and inverse is $(-x,-a^{-x}u)$; associativity follows by expanding both bracketings. Projection onto the first coordinate is a group homomorphism with normal cyclic kernel of order $p$. Under the change $U=a^{-x}u$, multiplication becomes $(x,U)(y,V)=(x+y,U+a^{-x}V)$, the semidirect product of the additive prime field by the cyclic group of order $p-1$. The group is abelian precisely when $a=1$; $a$ need not be a generator of the multiplicative field group.