Solution
= Solution
The class of finite groups is \b[not first-order axiomatizable]. If $S$ axiomatized it, add constants $c_1,c_2,\ldots$ and all inequalities $c_i\ne c_j$. Every finite fragment has a sufficiently large finite cyclic group as a model. The <compactness theorem> would then give an infinite group satisfying $S$, contradicting the definition of the class. This is the group instance of the fact that <finite structures are not first-order axiomatizable>.