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

Year 2024, Volume: 29 Issue: 1, 157 - 165, 30.04.2024

Ömer Küsmüş , İsmail Denizler , Richard M. Low

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.

Keywords

Devirli grup, Direkt çarpım grubu, Grup halkaları, İntegral grup halkaları, Kompleks temsil, Simetrik grup

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

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