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

Year 2020, Volume: 9 Issue: 2, 609 - 614, 15.06.2020

Ömer Küsmüş

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.

Keywords

Burulmalı birimsel elemanlar, İntegral grup halkaları, İki-devirli grup, Zassenhaus konjektürleri

On Torsion Units in Integral Group Ring of A Dicyclic Group

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.

Keywords

Torsion units, Integral group rings, Dicyclic Group, Zassenhaus Conjectures

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.