Is the category of groups with group-homomorphisms the same as the category of models of group theory with elementary maps? 


If not so: why?



Category of groups = Category of models of group theory?
The Categories will be fundamentally different. The category of groups with group homomorphisms, (even with monomorphisms) enjoys a directedness property: any two groups can map monomorphically into their direct sum. 


But in the category of models of group theory under elementary maps, the finite groups map elementarily only into isomorphic copies of themselves. The size of any particular finite group (or indeed any model in any theory) is first order expressible, and any elementary map will be an isomorphism.


