Mathematica Bohemica, Vol. 146, No. 4, pp. 429-455, 2021
On unit group of finite semisimple group algebras of non-metabelian groups up to order 72
Gaurav Mittal, Rajendra Kumar Sharma
Received August 8, 2019. Published online January 18, 2021.
Abstract: We characterize the unit group of semisimple group algebras $\mathbb{F}_qG$ of some non-metabelian groups, where $F_q$ is a field with $q=p^k$ elements for $p$ prime and a positive integer $k$. In particular, we consider all 6 non-metabelian groups of order 48, the only non-metabelian group $((C_3\times C_3)\rtimes C_3)\rtimes C_2$ of order 54, and 7 non-metabelian groups of order 72. This completes the study of unit groups of semisimple group algebras for groups upto order 72.
Keywords: unit group; finite field; Wedderburn decomposition
Affiliations: Gaurav Mittal, Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India, email: gmittal@ma.iitr.ac.in; Rajendra Kumar Sharma, Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India, e-mail: rksharma@maths.iitd.ac.in