TY - JOUR

T1 - Normal subgroups of nonstandard symmetric and alternating groups

AU - Kaye, Richard

AU - Allsup, John

PY - 2007/1/31

Y1 - 2007/1/31

N2 - Let M be a nonstandard model of Peano Arithmetic with domain M and let n is an element of M be nonstandard. We study the symmetric and alternating groups S-n and A(n) 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 R-valued metric on S-n=S-n/B-S and A(n)=A(n)/B-A (where B-S, B-A are the maximal normal subgroups of Sn and An identified earlier) making these groups into topological groups, and by showing that if M is N-1-saturated then S-n and A(n) are complete with respect to this metric.

AB - Let M be a nonstandard model of Peano Arithmetic with domain M and let n is an element of M be nonstandard. We study the symmetric and alternating groups S-n and A(n) 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 R-valued metric on S-n=S-n/B-S and A(n)=A(n)/B-A (where B-S, B-A are the maximal normal subgroups of Sn and An identified earlier) making these groups into topological groups, and by showing that if M is N-1-saturated then S-n and A(n) are complete with respect to this metric.

UR - http://www.scopus.com/inward/record.url?scp=33846841164&partnerID=8YFLogxK

U2 - 10.1007/s00153-006-0030-2

DO - 10.1007/s00153-006-0030-2

M3 - Article

SN - 1432-0665

SN - 1432-0665

VL - 46

SP - 107

EP - 121

JO - Archive for Mathematical Logic

JF - Archive for Mathematical Logic

IS - 2

ER -