Invariance and definability, with and without equality

The dual character of invariance under transformations and definability by some operations has been used in classical work by for example Galois and Klein. Following Tarski, philosophers of logic have claimed that logical notions themselves could be characterized in terms of invariance. In this paper, we generalize a correspondence due to Krasner between invariance under groups of permutations and definability in $\La_{\infty\infty}$ so as to cover the cases (quantifiers, logics without equality) that are of interest in the logicality debates, getting McGee's theorem about quantifiers invariant under all permutations and definability in pure $\La_{\infty\infty}$ as a particular case. We also prove some optimality results along the way, regarding the kind of relations which are needed so that every subgroup of the full permutation group is characterizable as a group of automorphisms.