On Distinguishing Quotients of Symmetric Groups

A study is carried out of the elementary theory of quotients of symmetric groups in a similar spirit to [10]. 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 Sκ(μ) = the subgroup consisting of elements whose support has cardinality < κ, for some κ ≤ μ. A many-sorted structure Mκλμ is defined which, it is shown, encapsulates the first order properties of the group Sλ(μ)/Sκ(μ). Specifically, these two structures are (uniformly) bi-interpretable, where the interpretation of Mκλμ in Sλ(μ)/Sκ(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) > 20 , cf(κ) ≤ 20 < κ,א0 < κ < 2 א0 , and κ = א0, we make a further analysis of the first order theory of Sλ(μ)/Sκ(μ), introducing many-sorted second order structures N 2 κλμ, all of whose sorts have cardinality at most 20 , and in terms of which we can completely characterize the elementary theory of the groups Sλ(μ)/Sκ(μ).