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.
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Baer R. (1935). Die Kompositionsreihe der Gruppe alle eieindeutigen Abbildungen einer unendlichen Menge auf sich. Studia. Math. 5: 15–17
Bhattacharjee, M., Macpherson, D., Möller, R.G., Neumann, P.M.: Notes on infinite permutation groups. In: Texts and Readings in Mathematics, vol. 12. Lecture Notes in Mathematics, 1698. Hindustan Book Agency, New Delhi (1997)
Bigelow S. (1998). Supplements of bounded permutation groups. J. Symbolic Logic 63(1): 89–102
Cameron P.J. (1990) Oligomorphic permutation groups. In: London Mathematical Society Lecture Note Series, vol. 152. Cambridge University Press, Cambridge (1990)
Kaye, R.W.: Models of Peano Arithmetic. In: Studies in Logic and the Foundations of Mathematics, vol. 15. Oxford University Press, Oxford (1989)
Kirby, L.A.S., Paris, J.B.: Initial segments of models of Peano’s axioms. In: Set Theory and Hierarchy Theory, V (Proceedings of the 3rd Conference, Bierutowice, 1976), pp. 211–226. Lecture Notes in Mathematics, vol. 619. Springer, Berlin (1977)
Macpherson H.D. and Neumann P.M. (1990). Subgroups of infinite symmetric groups. J. Lond. Math. Soc. 42(1): 64–84
Schreier J. and Ulam S. (1933). Über die permutatinosgrupe der natürlichen zahlenfolge. Studia Math. 4: 134–141
Wielandt H. (1964). Finite Permutation Groups (Translated from the German by R. Bercov). Academic Press, New York
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Allsup, J., Kaye, R. Normal subgroups of nonstandard symmetric and alternating groups. Arch. Math. Logic 46, 107–121 (2007). https://doi.org/10.1007/s00153-006-0030-2
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DOI: https://doi.org/10.1007/s00153-006-0030-2