Mathematica Bohemica, Vol. 148, No. 4, pp. 631-646, 2023
The unit groups of semisimple group algebras of some non-metabelian groups of order 144
Gaurav Mittal, Rajendra Kumar Sharma
Received May 10, 2022. Published online December 19, 2022.
Abstract: We consider all the non-metabelian groups $G$ of order 144 that have exponent either 36 or 72 and deduce the unit group $U(\mathbb{F}_qG)$ of semisimple group algebra $\mathbb{F}_qG$. Here, $q$ denotes the power of a prime, i.e., $q=p^r$ for $p$ prime and a positive integer $r$. Up to isomorphism, there are 6 groups of order 144 that have exponent either 36 or 72. Additionally, we also discuss how to simply obtain the unit groups of the semisimple group algebras of those non-metabelian groups of order 144 that are a direct product of two nontrivial groups. In all, this paper covers the unit groups of semisimple group algebras of 17 non-metabelian groups.
Keywords: unit group; finite field; Wedderburn decomposition
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Affiliations: Gaurav Mittal (corresponding author), Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee - 247 667, Uttarakhand, India, e-mail: gmittal@ma.iitr.ac.in; Rajendra Kumar Sharma, Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi - 110016, India, e-mail: rksharmaiitd@gmail.com
