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 show that any block of a group algebra of some finite group which is of wild representation type has many families of stable tubes.

Assume that S is a commutative complete discrete valuation domain of characteristic p, S* is the unit group of S and $G=G_p\times B$ is a finite group, where $G_p$ is a p-group and B is a p’-group. Denote by $S^{\lambda }G$ the twisted group algebra of G over S with a 2-cocycle λ ∈ Z²(G,S*). We give necessary and sufficient conditions for $S^{\lambda }G$ to be of OTP representation type, in the sense that every indecomposable $S^{\lambda }G$-module is isomorphic to the outer tensor product V W of an indecomposable $S^{\lambda }{G}_p$-module V and an irreducible $S^{\lambda }B$-module...

Let $\mathbb{Z}{̂}_p$ be the ring of p-adic integers, $U\left(\mathbb{Z}{̂}_p\right)$ the unit group of $\mathbb{Z}{̂}_p$ and $G=G_p\times B$ a finite group, where $G_p$ is a p-group and B is a p’-group. Denote by $\mathbb{Z}{̂}_p^{\lambda }G$ the twisted group algebra of G over $\mathbb{Z}{̂}_p$ with a 2-cocycle $\lambda \in Z²(G,U\left(\mathbb{Z}{̂}_p\right))$. We give necessary and sufficient conditions for $\mathbb{Z}{̂}_p^{\lambda }G$ to be of OTP representation type, in the sense that every indecomposable $\mathbb{Z}{̂}_p^{\lambda }G$-module is isomorphic to the outer tensor product V W of an indecomposable $\mathbb{Z}{̂}_p^{\lambda }{G}_p$-module V and an irreducible $\mathbb{Z}{̂}_p^{\lambda }B$-module W.

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.

For a finite group $G$ and a non-linear irreducible complex character $\chi$ of $G$ write $\upsilon \left(\chi \right)=\{g\in G\mid \chi (g)=0\}$. In this paper, we study the finite non-solvable groups $G$ such that $\upsilon \left(\chi \right)$ consists of at most two conjugacy classes for all but one of the non-linear irreducible characters $\chi$ of $G$. In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable $\varphi$-groups. As a corollary, we answer Research Problem $2$ in [Y. Berkovich and L. Kazarin: Finite...