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Almost Locally Free Groups and a Theorem of Magnus: Some Questions

Almost Locally Free Groups and a Theorem of Magnus: Some Questions Ben Fine observed that a theorem of Magnus on normal closures of elements in free groups is first order expressible and thus holds in every elementarily free group. This classical theorem, vintage 1931, asserts that if two elements in a free group have the same normal closure, then either they are conjugate or one is conjugate to the inverse of the other in the free group. An examination of a set of sentences capturing this theorem reveals that the sentences are universal-existential. Consequently the theorem holds in the almost locally free groups of Gaglione and Spellman. We give a sufficient condition for the theorem to hold in every fully residually free group as well as a sufficient condition for the theorem to hold, even more generally, in every residually free group. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups - Complexity - Cryptology de Gruyter

Almost Locally Free Groups and a Theorem of Magnus: Some Questions

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Publisher
de Gruyter
Copyright
© Heldermann Verlag
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/GCC.2009.181
Publisher site
See Article on Publisher Site

Abstract

Ben Fine observed that a theorem of Magnus on normal closures of elements in free groups is first order expressible and thus holds in every elementarily free group. This classical theorem, vintage 1931, asserts that if two elements in a free group have the same normal closure, then either they are conjugate or one is conjugate to the inverse of the other in the free group. An examination of a set of sentences capturing this theorem reveals that the sentences are universal-existential. Consequently the theorem holds in the almost locally free groups of Gaglione and Spellman. We give a sufficient condition for the theorem to hold in every fully residually free group as well as a sufficient condition for the theorem to hold, even more generally, in every residually free group.

Journal

Groups - Complexity - Cryptologyde Gruyter

Published: Oct 1, 2009

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