Research Article

On the Unit Group of the Integral Group Ring Z(S_3×C_3)

Volume: 29 Number: 1 April 30, 2024
Ömer Küsmüş *, İsmail Denizler , Richard M. Low
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On the Unit Group of the Integral Group Ring Z(S_3×C_3)

Abstract

Describing the group of units in the integral group ring is a famous and classical open problem. Let S_3 and C_3 be the symmetric group of order 6 and a cyclic group of order 3, respectively. In this paper, a description of the units of the integral group ring Z(S_3×C_3) of the direct product group S_3×C_3 concerning a complex representation of degree two is given. As a result, a part of the conjecture which is introduced in (Low, 2008) and related to group rings over a complex integral domain is resolved using representation theory.

Keywords

Complex representation , Cyclic group , Direct product group , Group rings , Integral group rings , Symmetric group

References

  1. Bilgin, T., Küsmüş, Ö., & Low, R. M. (2016). A characterization of the unit group in Z[T×C_2]. Bulletin of the Korean Mathematical Society, 53(4), 1105-1112. doi:10.4134/BKMS.b150526
  2. Eisele, F., Kiefer, A., & Gelder, I. V. (2015). Describing units of integral group rings up to commensurability. Journal of Pure and Applied Algebra, 219(7), 2901-2916. doi:10.1016/j.jpaa.2014.09.031
  3. Hanoymak, T., & Küsmüş, Ö. (2023). A generalization of G-nilpotent units in commutative group rings to direct product groups. Yuzuncu Yil University Journal of the Institute of Natural and Applied Sciences, 28(1), 8-18. doi:10.53433/yyufbed.1097581
  4. Jespers, E., & Parmenter, M. M. (1992). Bicyclic units in ZS_3. Bulletin of the Belgium Mathematical Society, 44, 141-146.
  5. Jespers, E., & Parmenter, M. M. (1993). Units of group rings of groups of order 16. Glasgow Mathematical Journal, 35, 367-379. doi:10.1017/S0017089500009952
  6. Jespers, E. (1995). Bicyclic units in some integral group rings. Canadian Mathematics Bulletin, 38(1), 80-86. doi:10.4153/CMB-1995-010-4
  7. Jespers, E., & del Rio, A. (2016). Group Ring Groups. Vol. 1 and 2. Berlin, Germany: De Gruyter.
  8. Kelebek, I. G., & Bilgin, T. (2014). Characterization of U_1 (Z[C_n×K_4]). European Journal of Pure and Applied Mathematics, 7(4), 462-471.
  9. Küsmüş, Ö. (2020). On idempotent units in commutative group rings. Sakarya University Journal of Science, 24(4), 782-790. doi:10.16984/saufenbilder.733935
  10. Low, R. M. (1998). Units in integral group rings for direct products. (PhD), Western Michigan University, Kalamazoo, MI.

Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

Publication Date

April 30, 2024

Submission Date

September 18, 2023

Acceptance Date

December 18, 2023

Published in Issue

Year 2024 Volume: 29 Number: 1

DOI

https://doi.org/10.53433/yyufbed.1361776

IZ

https://izlik.org/JA28ZE97XY

Cite

RIS / Bibtex
APA
Küsmüş, Ö., Denizler, İ., & Low, R. M. (2024). On the Unit Group of the Integral Group Ring Z(S_3×C_3). Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 29(1), 157-165. https://doi.org/10.53433/yyufbed.1361776