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Keywords:
unit group; finite field; Wedderburn decomposition
unit group; finite field; Wedderburn decomposition
Summary:
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.\looseness -1
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