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Composites of translations and odd rational powers act freely

Published online by Cambridge University Press:  17 April 2009

Stephen D. Cohen  and
A.M.W. Glass
Stephen D. Cohen
Affiliation:
Department of MathematicsUniversity of GlasgowGlasgow G12 8QW, Scotland
A.M.W. Glass
Affiliation:
Department of Mathematics and StatisticsBowling Green State UniversityBowling Green OH 43403-0221, United States of America
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Abstract

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It is shown that no non-trivial composition of translations x ↦ x + a and odd rational powers x ↦p/q, where p,q are odd co-prime integers, positive or negative with p/q≠±, acts like the identity on a field of characteristic zero. This extends a theorem of Adeleke, Glass, and Morley in which only odd positive rational powers were considered. Moreover, the nature of the proof itself (by field theory) is a simplification and natural refinement of previous proofs. It has applications in other settings.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 51 , Issue 1 , February 1995 , pp. 73 - 81
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Adeleke, S.A., Glass, A.M.W. and Morley, L., ‘Arithmetic permutations’, J. London Moth. Soc. 43 (1991), 255–268.CrossRefGoogle Scholar
[2]Cohen, S.D., ‘The group of translations and positive rational powers is free’, Quart. J. Math. Oxford (to appear).Google Scholar
[3]Garling, D.J.H., A course in Galois theory (Cambridge University Press, Cambridge, 1986).Google Scholar
[4]Lyndon, R.C. and Ullman, J.L., ‘Groups generated by two parabolic linear fractional transformations’, Canad. J. Math. 21 (1969), 1388–1403.CrossRefGoogle Scholar
[5]White, S., ‘The group generated by x ↦ x+1 and x ↦ x p is free’, J. Algebra 118 (1988), 408–422.CrossRefGoogle Scholar