On distinguishing quotients of symmetric groups Original Research Article

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 Sκ(μ) = the subgroup consisting of elements whose support has cardinality < κ, for some κ ⩽ μ. A many-sorted structure View the MathML source, is defined which, it is shown, encapsulates the first order properties of the group View the MathML source Specifically, these two structures are (uniformly) bi-interpretable. where the interpretation of View the MathML source, in View the MathML source 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 View the MathML source, and View the MathML source, we make a further analysis of the first order theory of View the MathML source. introducing many-sorted second order structures View the MathML source, all of whose sorts have cardinality at most View the MathML source, and in terms of which we can completely characterize the elementary theory of the groups View the MathML source.