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We study the inverse problem of the determination of a group algebra from the knowledge of its Wedderburn decomposition. We show that a certain class of matrix rings always occur as summands of finite group algebras.

We characterize the unit group of semisimple group algebras ${𝔽}_{q}G$ 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 $((C_3\times C_3)⋊C_3)⋊C_2$ 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.

We consider all the non-metabelian groups $G$ of order $144$ that have exponent either $36$ or $72$ and deduce the unit group $U\left({𝔽}_{q}G\right)$ of semisimple group algebra ${𝔽}_{q}G$. 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...

We give the characterization of the unit group of ${𝔽}_{q}SL(2,{\mathbb{Z}}_5)$, where ${𝔽}_q$ is a finite field with $q=p^k$ elements for prime $p>5,$ and $SL(2,{\mathbb{Z}}_5)$ denotes the special linear group of $2\times 2$ matrices having determinant $1$ over the cyclic group ${\mathbb{Z}}_5$.

We characterize the unit group for the group algebras of non-metabelian groups of order 128 over the finite fields whose characteristic does not divide the order of the group. Up to isomorphism, there are 2328 groups of order 128 and only 14 of them are non-metabelian. We determine the Wedderburn decomposition of the group algebras of these non-metabelian groups and subsequently characterize their unit groups.