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İki-devirli Bir Grubun Integral Grup Halkasındaki Burulmalı Birimsel Elemanlar Üzerine

Year 2020, , 609 - 614, 15.06.2020
https://doi.org/10.17798/bitlisfen.624068

Abstract

Bu
makalede temel amaç,
 iki-devirli grubunun  integral grup halkasındaki burulmalı birimsel
elemanların yapısını, ikinci dereceden bir kompleks indirgenemez güvenilir
temsil kullanarak karakterize etmektir. Birinci Zassenhaus konjektürü (ZC1) ile
göstereceğiz ki
 integral grup halkasındaki aşikar olmayan
burulmalı birimsel elemanlar 3, 4 veya 6 mertebeli ve bunların her biri üç
serbest parametre cinsinden ifade edilebilir.

References

  • Bächle A. Herman A. Konovalov A. Margolis L. Singh G. 2018. The Status of the Zassenhaus Conjecture for Small Groups, Experimental Mathematics, 27: 431-436.
  • Herman A. Singh G. 2015.Revisiting the Zassenhaus Conjecture on Torsion Units for the Integral Group Rings of Small Groups, Proc. Math. Sci., 125(2): 167-172. Bhandari A. K. Luthar I. S. 1993. Torsion Units of Integral Group Rings of Metacyclic Groups, J. Number Theory, 17: 170-183.
  • Milies C. P. Ritter J. Sehgal S. K. 1986. On A Conjecture of Zassenhaus on Torsion Units in Integral Group Rings II, Proc. Amer. Math. Soc. 97: 201-206.
  • Games D. G. Liebeck M. W. 1986. Representation and Characters of Groups, Cambridge University Press.
  • Hughes I. Pearson K. R. 1972. The Group of Units of the Integral Group Ring ZS_3, Canad. Math. Bull. 15: 529-534.
  • Gildea J. 2013. Zassenhaus Conjecture for Integral Group Rings of Simple Linear Groups, J. Algebra Appl. 12(6).
  • Ari K. 2003. On Torsion Units in the Group Ring ZA_4 and the First Conjecture of Zassenhaus, Int. Math. J. 9(3): 953-958.
  • Caecido M. Margolis L. del Rio A. 2013 Zassenhaus Conjecture for Cyclic-by-Abelian Groups, J. London Math. Soc. 88: 65-78.
  • Hertweck M, 2002. Another Counterexample to a Conjecture of Zassenhaus, Contributions to Algebra and Geometry, 43: 513-520.
  • Hertweck M. 2008. Zassenhaus Conjecture for A_6, Proc. Indian Acad. Sci. (Math. Sci.), 118: 189-195.
  • Hertweck M. 2008. On Torsion Units in Integral Group Rings of Certain Metabelian Groups, Proc. Edinb. Math. Soc. 51: 363-385.
  • Allen P. J. Hobby C. 1987. A Note on The Unit Group of ZS_3, Proc. Amer. Math. Soc. 99: 9-14.
  • Jespers E, Parmenter M. M. 1992. Bicyclic Units in ZS_3, Bull. Belg. Math. Soc., 44: 141-146.
  • Eisele F. Margolis L. 2018. A Counterexample to the first Zassanhaus Conjecture, Advances in Mathematics, 339: 599-641.
  • Sehgal S. K. 1993. Units in Integral Group Rings, Marcel Dekker, New York, Basel.
  • Bilgin T. 2004. Parametrization of Torsion Units in U1(ZS_3), Math. Comput. Appl., 9: 73-77.
  • Bilgin T. 2004. Parametrization of Torsion Units in U1(ZD_4), Int. J. Math. Game Theory Algebra, 14: 83-87.
  • Bilgin T. Ari K. 2007. Parametrization of Torsion Units in U1(ZD_5), Int. J. Algebra, 1: 347-352.

On Torsion Units in Integral Group Ring of A Dicyclic Group

Year 2020, , 609 - 614, 15.06.2020
https://doi.org/10.17798/bitlisfen.624068

Abstract

In
this paper, the main aim is to characterize the structure of torsion units in
integral group ring
 of dicyclic group by
using a complex faithful irreducible representation of degree 2. We show by the
first of Zassenhaus conjectures (ZC1) that non-trivial torsion units in
 are of order 3, 4 or 6 and each of them can be
expressed in terms of 3 free parameters.

References

  • Bächle A. Herman A. Konovalov A. Margolis L. Singh G. 2018. The Status of the Zassenhaus Conjecture for Small Groups, Experimental Mathematics, 27: 431-436.
  • Herman A. Singh G. 2015.Revisiting the Zassenhaus Conjecture on Torsion Units for the Integral Group Rings of Small Groups, Proc. Math. Sci., 125(2): 167-172. Bhandari A. K. Luthar I. S. 1993. Torsion Units of Integral Group Rings of Metacyclic Groups, J. Number Theory, 17: 170-183.
  • Milies C. P. Ritter J. Sehgal S. K. 1986. On A Conjecture of Zassenhaus on Torsion Units in Integral Group Rings II, Proc. Amer. Math. Soc. 97: 201-206.
  • Games D. G. Liebeck M. W. 1986. Representation and Characters of Groups, Cambridge University Press.
  • Hughes I. Pearson K. R. 1972. The Group of Units of the Integral Group Ring ZS_3, Canad. Math. Bull. 15: 529-534.
  • Gildea J. 2013. Zassenhaus Conjecture for Integral Group Rings of Simple Linear Groups, J. Algebra Appl. 12(6).
  • Ari K. 2003. On Torsion Units in the Group Ring ZA_4 and the First Conjecture of Zassenhaus, Int. Math. J. 9(3): 953-958.
  • Caecido M. Margolis L. del Rio A. 2013 Zassenhaus Conjecture for Cyclic-by-Abelian Groups, J. London Math. Soc. 88: 65-78.
  • Hertweck M, 2002. Another Counterexample to a Conjecture of Zassenhaus, Contributions to Algebra and Geometry, 43: 513-520.
  • Hertweck M. 2008. Zassenhaus Conjecture for A_6, Proc. Indian Acad. Sci. (Math. Sci.), 118: 189-195.
  • Hertweck M. 2008. On Torsion Units in Integral Group Rings of Certain Metabelian Groups, Proc. Edinb. Math. Soc. 51: 363-385.
  • Allen P. J. Hobby C. 1987. A Note on The Unit Group of ZS_3, Proc. Amer. Math. Soc. 99: 9-14.
  • Jespers E, Parmenter M. M. 1992. Bicyclic Units in ZS_3, Bull. Belg. Math. Soc., 44: 141-146.
  • Eisele F. Margolis L. 2018. A Counterexample to the first Zassanhaus Conjecture, Advances in Mathematics, 339: 599-641.
  • Sehgal S. K. 1993. Units in Integral Group Rings, Marcel Dekker, New York, Basel.
  • Bilgin T. 2004. Parametrization of Torsion Units in U1(ZS_3), Math. Comput. Appl., 9: 73-77.
  • Bilgin T. 2004. Parametrization of Torsion Units in U1(ZD_4), Int. J. Math. Game Theory Algebra, 14: 83-87.
  • Bilgin T. Ari K. 2007. Parametrization of Torsion Units in U1(ZD_5), Int. J. Algebra, 1: 347-352.
There are 18 citations in total.

Details

Primary Language English
Journal Section Araştırma Makalesi
Authors

Ömer Küsmüş 0000-0001-7397-0735

Publication Date June 15, 2020
Submission Date September 24, 2019
Acceptance Date April 8, 2020
Published in Issue Year 2020

Cite

IEEE Ö. Küsmüş, “On Torsion Units in Integral Group Ring of A Dicyclic Group”, Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, vol. 9, no. 2, pp. 609–614, 2020, doi: 10.17798/bitlisfen.624068.



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