Is division axiomatizable? Consider a set $G$ with a group operation. We can define a division operation $a*(b^{-1})$ and call it $\operatorname{div}$. Is the class of division operations first order axiomatizable? And if so, is it finitely axiomatizable?

Let $\star$ be your operator. On a group, this can be axiomatized as:
$$\forall x(1\star(1\star x)=x)\text{ (A)}\\ \forall x(x\star x = 1)\text{ (B)}\\\forall x,y,z\left((x\star y)\star z = x\star(z\star(1\star y))\right)\text{ (C)}$$
We can quickly show:
$$\begin{align}
x\star 1 &= (1\star(1\star x))\star 1 \text{ (A)}\\
&=1\star(1\star(1\star(1\star x)))\text{ (C)}\\
&=1\star(1\star x)\text{ (A)}\\
&=x\text{ (A)}
\end{align}
$$
Then if you define $x^{-1}=1\star x$ and $x\cdot y = x\star(1\star y)$, we can show:
$$x\cdot x^{-1} = x\star(1\star(1\star x)) = x\star x = 1\\
x^{-1}\cdot x = (1\star x)\star (1\star x)=1\\
1\cdot x = 1\star(1\star x)=x\\
x\cdot 1 = x\star (1\star 1) = x\star 1 = 1\\
\begin{align}(x\cdot y)\cdot z&=(x\star(1\star y))\star(1\star z)\\
&=x\star\left((1\star z)\star(1\star(1\star y))\right)\\
&=x\star\left((1\star z)\star y\right)\\
&=x\star(1\star (y\star(1\star z)))\\
&= x\cdot(y\cdot z)
\end{align}
$$
and finally:
$$a\cdot b^{-1} = a\star(1\star(1\star b)) = a\star b$$
So you've got all your group axioms.
