I'm trying to prove that the class of all finite groups is not finitely axiomatizable. I was trying to do this in the same way one proves that the class of finite sets is not axiomatizable, i.e.
Let $\phi_n=\forall x_1\ldots\forall x_n\left[\bigwedge\limits_{1\leq i<j\leq n}\neg(x_i=x_j)\wedge\forall x_{n+1}\bigvee\limits_{1\leq i\leq n} x_{n+1}=x_i\right]$
Then we want to express $\phi_1\vee\phi_2\vee\ldots$
But there is no formula $\psi$ such that $(V)\models\psi\iff V$ is finite. Since: Assume there is a $\psi$. Find the quantifier rank $n$ of $\psi$. Then $(\{0,\ldots,n-1\})\equiv_n(\{0,\ldots,n\})$ but not $(\{0,\ldots,n-1\})\equiv_{n+1}(\{0,\ldots,n\})$ and therefore, for all $i\geq n$:
if $(\{0,\ldots,i\})\models\psi$ then $(\mathbb{N})\models\psi$.
But can this be done for finite groups as well, since we are dealing with multiplication?