Abstract
We solve Hilbert’s fifth problem for local groups: every locally euclidean local group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957, but this proof is seriously flawed. We use methods from nonstandard analysis and model our solution after a treatment of Hilbert’s fifth problem for global groups by Hirschfeld.
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@article {B, MRKEY = {900105},
AUTHOR = {Brown, D. R. and Houston, R. S.},
TITLE = {Cancellative semigroups on manifolds},
JOURNAL = {Semigroup Forum},
FJOURNAL = {Semigroup Forum},
VOLUME = {35},
YEAR = {1987},
NUMBER = {3},
PAGES = {279--302},
ISSN = {0037-1912},
CODEN = {SMGFAN},
MRCLASS = {22A20},
MRNUMBER = {89a:22005},
MRREVIEWER = {Haskell Cohen},
ZBLNUMBER = {0626.22001},
} -
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@book {D, MRKEY = {0505473},
AUTHOR = {Davis, Martin},
TITLE = {Applied Nonstandard Snalysis},
PUBLISHER = {Wiley-Interscience [John Wiley \& Sons]},
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ISBN = {0-471-19897-8},
MRCLASS = {02H25},
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ZBLNUMBER = {0359.02060},
} -
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@misc{vdD,
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TITLE={unpublished notes},
} -
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AUTHOR = {Gleason, Andrew M.},
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JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
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[He] Foundations of Nonstandard Analysis: A Gentle Introduction to Nonstandard Extensions; Nonstandard Analysis: Theory and ApplicationsDordrecht: Kluwer Academic Publishers Group, 1997.
@proceedings {He, TITLE = {Foundations of Nonstandard Analysis: A Gentle Introduction to Nonstandard Extensions; Nonstandard Analysis: Theory and Applications},
SERIES = {{\rm NATO} Adv. Sc. Inst. Series {\rm C}},
NUMBER = {493},
BOOKTITLE = {Proc. {{\rm NATO}} {A}dvanced {S}tudy {I}nstitute on {N}onstandard {A}nalysis and its {A}pplications},
VENUE={{E}dinburgh, {J}une 30--{J}uly 13, 1996},
EDITOR={Leif O. Arkeryd, Nigel J. Cutland and C. Ward Henson},
PUBLISHER = {Kluwer Academic Publishers Group},
ADDRESS = {Dordrecht},
YEAR = {1997},
PAGES = {xiv+366},
ISBN = {0-7923-4586-X},
MRCLASS = {03-06 (00B25 03H05)},
MRNUMBER = {98i:03006},
ZBLNUMBER={0910.03040},
} -
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@article {H, MRKEY = {967314},
AUTHOR = {Hirschfeld, Joram},
TITLE = {The nonstandard treatment of {H}ilbert's fifth problem},
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FJOURNAL = {Transactions of the American Mathematical Society},
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ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {22E15 (03H05 22D05 46Q05)},
MRNUMBER = {90m:22021},
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DOI = {10.2307/2001608},
ZBLNUMBER = {0705.03033},
} -
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@article {Ho, MRKEY = {929446},
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VOLUME = {37},
YEAR = {1988},
NUMBER = {1},
PAGES = {93--111},
ISSN = {0037-1912},
CODEN = {SMGFAN},
MRCLASS = {22A15 (22E99)},
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ZBLNUMBER = {0635.22003},
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@article {J, MRKEY = {0089997},
AUTHOR = {Jacoby, Robb},
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JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
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ISSN = {0003-486X},
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MRNUMBER = {19,751c},
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} -
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@book {Ka, MRKEY = {0276398},
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@misc{Mc,
author={McGaffey, T.},
TITLE={A partial solution to a generalization of Hilbert's local fifth problem: the standard part of a locally euclidean local nonstandard Lie group is an analytic Lie group},
MARGINNOTE={Author: Please provide more information for this entry},
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@book {MZ, MRKEY = {0073104},
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MRCLASS = {22E05 (58H05)},
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} -
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@techreport{Pl,
author={Plaut, C.},
TITLE={Associativity and the local version of Hilbert's fifth problem},
TYPE={notes},
INSTITUTION={University of Tenessee},
YEAR={1993},
} -
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@book{P,
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ADDRESS={Princeton, NJ},
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} -
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@article {S, MRKEY = {0396840},
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ISSN = {0025-2611},
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MRNUMBER = {53 \#700},
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DOI = {10.1007/BF01170531},
ZBLNUMBER = {0328.22011},
}
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