I was playing with the Hermitian curve $y^q + y = x^{q+1}$ over the field $GF(q^2)$ and chanced upon the following (non Abelian) group law on the points of the affine curve:


$(a,b) \* (c,d) = (a+c,b+d+ac^q)$.


Over $GF(q^2)$ the group has $q^3$ points, the identity is $(0,0)$ and 
the inverse of $(a,b)$ is $(-a,b^q)$.


So my question is whether this is known. I could not find a reference.
If you haven't seen this before and if you're interested in seeing how I found this group law, just shoot me an email.


Thanks for your time.
Best,
Hiren



Non-abelian group from affine hermitian curve
Yes, that group law is known. It is a disguised form of a Sylow $p$-subgroup of the automorphism group of the Hermitian curve (namely $\text{PGU}\_3(q^2)$), where $p$ is the characteristic of $\mathbf{F}\_q$. This group is well understood, for instance it's an extraspecial $p$-group.


For any $\mathbf{F}\_{q^2}$-rational point $(a,b)$ on the curve $y^q+y=x^{q+1}$, the map
$$ (x,y) \mapsto (x+a, y+a^qx+b) $$
is an automorphism of the curve. It is known that the set of all such automorphisms forms a group $G$, and that this group $G$ acts faithfully and transitively on the set $S$ of $\mathbf{F}\_{q^2}$-rational points on the curve. Moreover, $G$ has order $q^3$, and is a Sylow $p$-subgroup of the automorphism group of the curve.


For any faithful transitive action of a group $G$ on a set $S$, we can pick a point $P\in S$ and identify the group element $g\in G$ with 
the point $g(P)\in S$. Then we obtain a group structure on $S$ via $g(P)\*h(P):=gh(P)$, where the identity element is $P$. Your group law on the set $S$ of $\mathbf{F}\_{q^2}$-rational points is obtained from the above group $G$ by putting $P=(0,0)$.


