Name of group of order $p-\bigl(\frac ap\bigr)$ constructed from field $\Bbb Z_p$?

Question

Let $p$ be an odd prime, and $a$ be an element of field $\Bbb Z_p$.

Define $l$ as the Legendre symbol $\displaystyle\biggl(\frac ap\biggr)$.

When $l=+1$, define $b$ as a particular solution of $b^2=a$ in $\Bbb Z_p$, e.g. the odd one in range $[1,p)$.

Define the set $\mathcal S$ as: $$\mathcal S=\begin{cases} \{\infty\}\cup\Bbb Z_p&\text{when }l=-1\\ \{\infty\}\cup\Bbb Z_p-\{0\}&\text{when }l=0\\ \{\infty\}\cup\Bbb Z_p-\{b,\,p-b\}&\text{when }l=+1 \end{cases}$$

Define internal law $\boxplus$ in $\mathcal S$ as: $$x\boxplus y=\begin{cases} y&\text{when }x=\infty\\ x&\text{when }x\ne\infty\text{ and }y=\infty\\ \infty&\text{when }x\ne\infty\text{ and }y=-x\\ (x+y)^{-1}(x\,y+a)&\text{(computed in $\Bbb Z_p$) otherwise} \end{cases}$$

$(S,\boxplus)$ is a finite Abelian group¹ of order $p-l$. What's its name in standard literature?

---

¹ The Abelian group axioms hold:

• Closure of $\boxplus$ follows from definitions, trivially for the case $l=-1$, and with some level of care in the other ones.
• The neutral is $\infty$. This follows from the first three cases of the definition of $\boxplus$.
• The opposite $-x$ is computed as in $\Bbb Z_p$ when $x\ne\infty$, and we define $-\infty$ as $\infty$. That $x\boxplus(-x)=\infty=(-x)\boxplus x$ follows from the first and third cases of the definition.

• Associativity $(x\boxplus y)\boxplus z=x\boxplus(y\boxplus z)$ needs to be proved in multiple cases:

  • When any of $x$, $y$ or $z$ is $\infty$, we use the first two cases of the definition.
  • Otherwise, when $y$ is $-x$ or $-z$, we use the first three cases of the definition.
  • In the remaining general case, it boils down to basic algebra. The exclusions in the definition of $\mathcal S$ prevent degenerate cases.
• Commutativity is easy.

Answer 1

For $k$ a field and $a\in k^*$ let $$H_{a,p} = \{ \pmatrix{r &x\\ -ax&r}\in GL_2(k), r^2+ax^2\ne 0\}$$ It is a subgroup of $GL_2(k)$ $$\pmatrix{r &x\\ -ax&r}\pmatrix{s&y\\ -ay&s}=\pmatrix{ -axy + rs & ry + sx\\-a( ry + sx)& -axy + rs }$$

Then $H_{a,p}/k^*$ is your subgroup of $PGL_2(k)$, which is isomorphic to the group of fractional linear transformations $$\pmatrix{r &x\\ -ax&r}\to \frac{tr+x}{-axt+r}= \cases{\frac{t+x/r}{-a\frac{x}rt+1}\ \text{if} \ r\ne 0\\ \frac{-1}{at}\text{ otherwise}}$$ That $a$ is a square or not tells if the subgroup of $GL_2(k)$ is a split or non-split torus. It would suggest the name "projective non-split torus" but it doesn't give any hit.

The case $a=0$ is degenerate and your group takes a completely different form, it is isomorphic to $(k,+)$ through $x\to \frac1x$.