Normal subgroups of nonstandard symmetric and alternating groups |
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Authors: | John Allsup Richard Kaye |
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Institution: | (1) School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK |
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Abstract: | Let be a nonstandard model of Peano Arithmetic with domain M and let be nonstandard. We study the symmetric and alternating groups S
n
and A
n
of permutations of the set internal to , 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 -valued metric on and (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 is -saturated then and are complete with respect to this metric.
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Keywords: | Models of arithmetic Nonstandard groups Permutation groups |
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