Öz
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.
Anahtar Kelimeler
Devirli grup Direkt çarpım grubu Grup halkaları İntegral grup halkaları Kompleks temsil Simetrik grup
Kaynakça
- 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.
Öz
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.
Anahtar Kelimeler
Complex representation Cyclic group Direct product group Group rings Integral group rings Symmetric group
Kaynakça
- 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.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Cebir ve Sayı Teorisi |
| Bölüm | Fen Bilimleri ve Matematik / Natural Sciences and Mathematics |
| Yazarlar | |
| Yayımlanma Tarihi | 30 Nisan 2024 |
| Gönderilme Tarihi | 18 Eylül 2023 |
| Yayımlandığı Sayı | Yıl 2024 Cilt: 29 Sayı: 1 |
