For any integer $k\geq 2,$ one can find a sentence $\lambda_k$ which translates 'there are at least $k$ things.' For example, $$\lambda_2=\exists v_1 \exists v_2 (v_1\neq v_2),$$ $$\lambda_3 = \exists v_1 \exists v_2 \exists v_3 (v_1\neq v_2 \wedge v_2\neq v_3 \wedge v_1\neq v_3),$$ and etc.
It is well known that the class of all infinite groups is an elementary class in the wider sense, denoted as $EC_\Delta,$ because it is a model of conjunction of the groups axioms together with $\{\lambda_2,\lambda_3,...\}$
Question: Instead of taking $\{\lambda_2,\lambda_3,...\},$ can we take conjunction of all $\lambda_2,\lambda_3,...$ so that the class of all infinite groups becomes an elementary class with a sentence being conjunction of all groups axioms and all $\lambda$'s?
First-order logic doesn't allow infinite conjunctions, so your proposed definition doesn't work. But this doesn't show that no such definition works, so the question "Is the class of infinite groups axiomatizable by a single sentence?" isn't answered by that.
However, it isn't hard to show that the answer is in fact "no," by compactness: if $\varphi$ is true in exactly the infinite groups, then $\neg\varphi$ is true in exactly the finite groups - but there are arbitrarily large finite groups ...