Normal subgroups of nonstandard symmetric and alternating groups

Abstract

Let $${\mathfrak{M}}$$ be a nonstandard model of Peano Arithmetic with domain M and let $${n \in M}$$ be nonstandard. We study the symmetric and alternating groups S n and A n of permutations of the set $${\{0,1,\ldots,n-1\}}$$ internal to $${\mathfrak{M}}$$ , and classify all their normal subgroups, identifying many externally defined such normal subgroups in the process. We provide evidence that A n and S n are not split extensions by these normal subgroups, by showing that any such complement if it exists, cannot be a limit of definable sets. We conclude by identifying an $${\mathbb{R}}$$ -valued metric on $${\tilde{S}_n = S_n /B_S}$$ and $${\tilde{A}_n = A_n /B_A}$$ (where B S , B A are the maximal normal subgroups of S n and A n identified earlier) making these groups into topological groups, and by showing that if $${\mathfrak{M}}$$ is $${\mathfrak\aleph_1}$$ -saturated then $${\tilde{S}_n}$$ and $${\tilde{A}_n}$$ are complete with respect to this metric.