Research Article
BibTex RIS Cite

PERMUTATIONS WITH A DISTINCT DIVISOR PROPERTY

Year 2021, , 1 - 14, 05.01.2021
https://doi.org/10.24330/ieja.851969

Abstract

A finite group of order $n$ is said to have the distinct divisor pro-perty (DDP) if there exists a permutation $g_1,\ldots, g_n$ of its elements such that $g_i^{-1}g_{i+1} \neq g_j^{-1}g_{j+1}$ for all $1\leq i<j<n$. We show that an abelian group is DDP if and only if it has a unique element of order 2. We also describe a construction of DDP groups via group extensions by abelian groups and show that there exist infinitely many non abelian DDP groups.

References

  • L. M. Batten and S. Sane, Permutations with a distinct difference property, Discrete Math., 261 (2003), 59-67.
  • S. Bauer-Mengelberg and M. Ferentz, On eleven-interval twelve-tone rows, Perspectives of New Music, 3 (1965), 93-103.
  • J. P. Costas, A study of a class of detection waveforms having nearly ideal range-Doppler ambiguity properties, Proceedings of the IEEE, 72 (1984), 996-1009.
  • J. P. Costas, Medium Constraints on Sonar Design and Performance, Tech. Rep. Class 1 Rep. R65EMH33, General Electric Company, Fairfield, CT, USA, 1965.
  • K. Drakakis, F. Iorio, S. Rickard and J. Walsh, Results of the enumeration of Costas arrays of order 29, Adv. Math. Commun., 5 (2011), 547-553.
  • H. Eimert, Lehrbuch der Zwolftontechnik, Weisbaden, Breitkopf und Hartel, 1952.
  • E. N. Gilbert, Latin squares which contain no repeated diagrams, SIAM R., 7(2) (1965), 189-198.
  • S. W. Golomb, Algebraic constructions for Costas arrays, J. Combin. Theory Ser. A, 37 (1984), 13-21.
  • S. W. Golomb, Construction of signals with favourable correlation properties, in: A. Pott et al. (Eds.), Difference Sets, Sequences and their Correlation Properties, Kluwer, Dordrecht, 1999, 159-194.
  • S. W. Golomb and H. Taylor, Constructions and properties of Costas arrays, Proceedings of the IEEE, 72(9) (1984), 1143-1163.
  • M. Gustar, Number of Difference Sets for Permutations of [2n] with Distinct Differences, The on-line encyclopedia of integer sequences, https://oeis. org/A141599, 2008.
  • F. H. Klein, Die Grenze der Halbtonwelt, Die Musik, 17 (1925), 281-286.
  • N. Slonimsky, Thesaurus of Scales and Melodic Patterns, Amsco Publications, 8th edition, New York, 1975.
Year 2021, , 1 - 14, 05.01.2021
https://doi.org/10.24330/ieja.851969

Abstract

References

  • L. M. Batten and S. Sane, Permutations with a distinct difference property, Discrete Math., 261 (2003), 59-67.
  • S. Bauer-Mengelberg and M. Ferentz, On eleven-interval twelve-tone rows, Perspectives of New Music, 3 (1965), 93-103.
  • J. P. Costas, A study of a class of detection waveforms having nearly ideal range-Doppler ambiguity properties, Proceedings of the IEEE, 72 (1984), 996-1009.
  • J. P. Costas, Medium Constraints on Sonar Design and Performance, Tech. Rep. Class 1 Rep. R65EMH33, General Electric Company, Fairfield, CT, USA, 1965.
  • K. Drakakis, F. Iorio, S. Rickard and J. Walsh, Results of the enumeration of Costas arrays of order 29, Adv. Math. Commun., 5 (2011), 547-553.
  • H. Eimert, Lehrbuch der Zwolftontechnik, Weisbaden, Breitkopf und Hartel, 1952.
  • E. N. Gilbert, Latin squares which contain no repeated diagrams, SIAM R., 7(2) (1965), 189-198.
  • S. W. Golomb, Algebraic constructions for Costas arrays, J. Combin. Theory Ser. A, 37 (1984), 13-21.
  • S. W. Golomb, Construction of signals with favourable correlation properties, in: A. Pott et al. (Eds.), Difference Sets, Sequences and their Correlation Properties, Kluwer, Dordrecht, 1999, 159-194.
  • S. W. Golomb and H. Taylor, Constructions and properties of Costas arrays, Proceedings of the IEEE, 72(9) (1984), 1143-1163.
  • M. Gustar, Number of Difference Sets for Permutations of [2n] with Distinct Differences, The on-line encyclopedia of integer sequences, https://oeis. org/A141599, 2008.
  • F. H. Klein, Die Grenze der Halbtonwelt, Die Musik, 17 (1925), 281-286.
  • N. Slonimsky, Thesaurus of Scales and Melodic Patterns, Amsco Publications, 8th edition, New York, 1975.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mohammad Javaherı This is me

Nikolai A. Krylov This is me

Publication Date January 5, 2021
Published in Issue Year 2021

Cite

APA Javaherı, M., & Krylov, N. A. (2021). PERMUTATIONS WITH A DISTINCT DIVISOR PROPERTY. International Electronic Journal of Algebra, 29(29), 1-14. https://doi.org/10.24330/ieja.851969
AMA Javaherı M, Krylov NA. PERMUTATIONS WITH A DISTINCT DIVISOR PROPERTY. IEJA. January 2021;29(29):1-14. doi:10.24330/ieja.851969
Chicago Javaherı, Mohammad, and Nikolai A. Krylov. “PERMUTATIONS WITH A DISTINCT DIVISOR PROPERTY”. International Electronic Journal of Algebra 29, no. 29 (January 2021): 1-14. https://doi.org/10.24330/ieja.851969.
EndNote Javaherı M, Krylov NA (January 1, 2021) PERMUTATIONS WITH A DISTINCT DIVISOR PROPERTY. International Electronic Journal of Algebra 29 29 1–14.
IEEE M. Javaherı and N. A. Krylov, “PERMUTATIONS WITH A DISTINCT DIVISOR PROPERTY”, IEJA, vol. 29, no. 29, pp. 1–14, 2021, doi: 10.24330/ieja.851969.
ISNAD Javaherı, Mohammad - Krylov, Nikolai A. “PERMUTATIONS WITH A DISTINCT DIVISOR PROPERTY”. International Electronic Journal of Algebra 29/29 (January 2021), 1-14. https://doi.org/10.24330/ieja.851969.
JAMA Javaherı M, Krylov NA. PERMUTATIONS WITH A DISTINCT DIVISOR PROPERTY. IEJA. 2021;29:1–14.
MLA Javaherı, Mohammad and Nikolai A. Krylov. “PERMUTATIONS WITH A DISTINCT DIVISOR PROPERTY”. International Electronic Journal of Algebra, vol. 29, no. 29, 2021, pp. 1-14, doi:10.24330/ieja.851969.
Vancouver Javaherı M, Krylov NA. PERMUTATIONS WITH A DISTINCT DIVISOR PROPERTY. IEJA. 2021;29(29):1-14.

Cited By