On distinguishing quotients of symmetric groups

A study of the elementary theory of quotients of symmetric groups is carried out in a similar spirit to Shelah (1973). Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S(μ) on an infinite cardinal μ are all of the form S_κ(μ)= the subgroup consisting of elements whose support has cardinality < κ, for some κ≤μ. A many-sorted structure M_κλμ is defined which, it is shown, encapsulates the first order properties of the group S_λ(μ)/S_κ(μ). Specifically, these two structures are (uniformly) bi-interpretable, where the interpretation of M_κλμ in S_λ(μ)/S_κ(μ) is in the usual sense, but in the other direction is in a weaker sense, which is nevertheless sufficient to transfer elementary equivalence. By considering separately the cases cf(κ)>2^Ν0, cf(κ)≤2^Ν0 <κ, Ν0 <κ <2^Ν0, and κ = Ν0, we make a further analysis of the first order theory of S_λ(μ)/S_κ(μ), introducing many-sorted second order structures script N² _κλμ, all of whose sorts have cardinality at most 2^Ν0, and in terms of which we can completely characterize the elementary theory of the groups S_λ(μ)/S_κ(μ).