Normal subgroups of nonstandard symmetric and alternating groups

John Allsup and Richard Kaye

School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK

Published in: Archive for Mathematical Logic, Springer Berlin / Heidelberg, ISSN 0933-5846 (Print) 1432-0665 (Online), Volume 46, Number 2 / February, 2007. DOI 10.1007/s00153-006-0030-2. Pages 107-121.

Received: 16 March 2006 Revised: 1 August 2006 Published online: 23 January 2007


Let 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 Sn and An of permutations of the set {0,1,...,n-1} internal to M, and classify all their normal subgroups, identifying many externally defined such normal subgroups in the process. We provide evidence that An and Sn 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 real-valued metric on S = Sn/BS and A = An/BA (where BS, BA are the maximal normal subgroups of Sn and An identified earlier) making these groups into topological groups, and by showing that if M is aleph-1 -saturated then S and A are complete with respect to this metric.

Keywords Models of arithmetic - Nonstandard groups - Permutation groups.


A follow-up paper is being prepared, in which many more of the properties of the group S = Sn/BS are explored. Note that it turns out that Sn/BS and An/BA are isomorphic, an easy fact that was overlooked in the published paper.

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