Normal subgroups of nonstandard symmetric and alternating groups

Archive for Mathematical Logic 46 (2):107-121 (2007)
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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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The arithmetic of cuts in models of arithmetic.Richard Kaye - 2013 - Mathematical Logic Quarterly 59 (4-5):332-351.

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References found in this work

Supplements of bounded permutation groups.Stephen Bigelow - 1998 - Journal of Symbolic Logic 63 (1):89-102.
Supplements of bounded permutation groups.Stephen Bigelow - 1998 - Journal of Symbolic Logic 63 (1):89-102.

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