Can a Group Be Proven Infinite with a Single Formula?

[sairalouise]

I'm trying to show that there is not one sentence (formula) that if a group satisfies this formula it is equivalent to the group being infinite. I can show this in a hap hazard way analogous to the same problem in the empty language , but how do you use the fact that the model is a group and there are arbitrarily large groups?

 

[AKG]

Let s be a sentence such that for all groups G, G models s iff G is infinite. Then a group G models ~s iff G is finite. So every finite group models ~s, and so

{~s} U {axioms of group theory}

has arbitrarily large finite models (since there are arbitrarily large finite groups). But a standard compactness argument yields that

{~s} U {axioms of group theory}

has an infinite model G which would be a group that models both s and ~s, contradiction.