= Twisted cyclic pair group
{title2=$(x,u)(y,v)=(x+y,a^yu+v)$}
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.
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