First order theory of permutation groups

We solve the problem of the elementary equivalence (definability) of the permutation groups over cardinals ℵ_α. We show that it suffices to solve the problem of elementary equivalence (definability) for the ordinals α in certain second order logic, and this is reduced to the case of α < (2^ℵ ₀)⁺. We solve a problem of Mycielski and McKenzie on embedding of free groups in permutation groups, and discuss some weak second-order quantifiers. An erratum to this article is available at .