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Z(S_3×C_3) İntegral Grup Halkasındaki Birimsel Elemanlar Grubu Üzerine

Year 2024, Volume: 29 Issue: 1, 157 - 165, 30.04.2024
https://doi.org/10.53433/yyufbed.1361776

Abstract

Verilen bir sonlu grubun integral grup halkasındaki birimsel elemanların grubunu belirlemek çoğu grup için meşhur ve klasik bir açık problemdir. S_3 ve C_3 sırasıyla 6 mertebeli simetrik grup ve 3 mertebeli bir devirli grup olsun. Bu makalede, iki dereceli bir kompleks temsile göre S_3×C_3 direkt çarpım grubunun Z(S_3×C_3) integral grup halkasının birimsel elemanlarının yapısı verilmektedir. Sonuç olarak, kompleks bir tamlık bölgesi üzerinde tanımlı grup halkalarına ilişkin (Low, 2008)’ de sunulan konjektürün bir kısmı, temsil teorisi kullanılarak çözülmüştür.

References

  • 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
  • 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
  • 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
  • Jespers, E., & Parmenter, M. M. (1992). Bicyclic units in ZS_3. Bulletin of the Belgium Mathematical Society, 44, 141-146.
  • 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
  • Jespers, E. (1995). Bicyclic units in some integral group rings. Canadian Mathematics Bulletin, 38(1), 80-86. doi:10.4153/CMB-1995-010-4
  • Jespers, E., & del Rio, A. (2016). Group Ring Groups. Vol. 1 and 2. Berlin, Germany: De Gruyter.
  • 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.
  • Küsmüş, Ö. (2020). On idempotent units in commutative group rings. Sakarya University Journal of Science, 24(4), 782-790. doi:10.16984/saufenbilder.733935
  • Low, R. M. (1998). Units in integral group rings for direct products. (PhD), Western Michigan University, Kalamazoo, MI.
  • Low, R. M. (2008). On the units of the integral group ring Z[G×C_p]. Journal of Algebra and its Applications, 7(3), 393-403. doi:10.1142/S0219498808002898
  • Milies, C. P., & Sehgal, S. K. (2002). An Introduction to Group Rings. London, UK: Kluwer Academic Publishers.
  • Sehgal, S. K. (1993). Units in Integral Group Rings. Essex, England: Longman Scientific & Technical.

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

Year 2024, Volume: 29 Issue: 1, 157 - 165, 30.04.2024
https://doi.org/10.53433/yyufbed.1361776

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.

References

  • 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
  • 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
  • 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
  • Jespers, E., & Parmenter, M. M. (1992). Bicyclic units in ZS_3. Bulletin of the Belgium Mathematical Society, 44, 141-146.
  • 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
  • Jespers, E. (1995). Bicyclic units in some integral group rings. Canadian Mathematics Bulletin, 38(1), 80-86. doi:10.4153/CMB-1995-010-4
  • Jespers, E., & del Rio, A. (2016). Group Ring Groups. Vol. 1 and 2. Berlin, Germany: De Gruyter.
  • 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.
  • Küsmüş, Ö. (2020). On idempotent units in commutative group rings. Sakarya University Journal of Science, 24(4), 782-790. doi:10.16984/saufenbilder.733935
  • Low, R. M. (1998). Units in integral group rings for direct products. (PhD), Western Michigan University, Kalamazoo, MI.
  • Low, R. M. (2008). On the units of the integral group ring Z[G×C_p]. Journal of Algebra and its Applications, 7(3), 393-403. doi:10.1142/S0219498808002898
  • Milies, C. P., & Sehgal, S. K. (2002). An Introduction to Group Rings. London, UK: Kluwer Academic Publishers.
  • Sehgal, S. K. (1993). Units in Integral Group Rings. Essex, England: Longman Scientific & Technical.
There are 13 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Natural Sciences and Mathematics / Fen Bilimleri ve Matematik
Authors

Ömer Küsmüş 0000-0001-7397-0735

İsmail Denizler 0000-0002-3360-6696

Richard M. Low 0000-0003-2320-8296

Publication Date April 30, 2024
Submission Date September 18, 2023
Published in Issue Year 2024 Volume: 29 Issue: 1

Cite

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