On unit group of finite semisimple group algebras of non-metabelian groups of order 108
On unit group of finite semisimple group algebras of non-metabelian groups of order 108
In this paper, we characterize the unit groups of semisimple group algebras $\mathbb{F}_qG$ of non-metabelian groups of order $108$, where $F_q$ is a field with $q=p^k$ elements for some prime $p > 3$ and positive integer $k$. Up to isomorphism, there are $45$ groups of order $108$ but only $4$ of them are non-metabelian. We consider all the non-metabelian groups of order $108$ and find the Wedderburn decomposition of their semisimple group algebras. And as a by-product obtain the unit groups.
Keywords:
Unit group Finite field, Wedderburn decomposition,
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- [1] A. Bovdi, J. Kurdics, Lie properties of the group algebra and the nilpotency class of the group of units, J. Algebra 212 (1999) 28–64.
- [2] V. Bovdi, M. Salim, On the unit group of a commutative group ring, Acta Sci. Math. (Szeged) 80 (2014) 433–445.
- [3] R. A. Ferraz, Simple components of the center of FG/J(FG), Comm. Algebra 36 (2008) 3191–3199.
- [4] B. Hurley, T. Hurley, Group ring cryptography, Int. J. Pure Appl. Math. 69 (2011) 67–86.
- [5] P. Hurley, T. Hurley, Codes from zero-divisors and units in group rings, Int. J. Inf. Coding Theory 1 (2009) 57–87.
- [6] G. Karpilovsky, The Jacobson radical of group algebras, Volume 135 Elsevier (1987).
- [7] M. Khan, R. K. Sharma, J. Srivastava, The unit group of FS4, Acta Math. Hungar. 118 (2008) 105-113.
- [8] R. Lidl, H. Niederreiter, Introduction to finite fields and their applications, Cambridge university press (1994).
- [9] S. Maheshwari, R. K. Sharma, The unit group of group algebra FqSL(2;Z3), J. Algebra Comb. Discrete Appl. 3 (2016) 1–6.
- [10] N. Makhijani, R. K. Sharma, J. Srivastava, A note on the structure of FpkA5=J(FpkA5), Acta Sci. Math. (Szeged) 82 (2016) 29–43.
- [11] C. P. Milies, S. K. Sehgal, An introduction to group rings, Springer Science & Business Media (2002).
- [12] G. Mittal, R. K. Sharma, On unit group of finite group algebras of non-metabelian groups up to order 72, Math Bohemica (2021)
- [13] G. Pazderski, The orders to which only belong metabelian groups, Math. Nachr. 95 (1980) 7–16.
- [14] S. Perlis, G. L. Walker, Abelian group algebras of finite order, Trans. Amer. Math. Soc. 68 (1950) 420–426.
- [15] R. K. Sharma, G. Mittal, On the unit group of a semisimple group algebra FqSL(2;Z5), Math Bohemica (2021).
- Başlangıç: 2015
- Yayıncı: İrfan ŞİAP
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