Let $G$ be a finite group $G$, $K$ a field of characteristic $p\ge 17$ and let $U$ be the group of units in $KG$. We show that if the derived length of $U$ does not exceed $4$, then $G$ must be abelian.

Let G be a finite p-group and let F be the field of p elements. It is shown that if G is elementary abelian-by-cyclic then the isomorphism type of G is determined by FG.

In this paper, we study the situation as to when the unit group U(KG) of a group algebra KG equals K*G(1 + J(KG)), where K is a field of characteristic p > 0 and G is a finite group.

In this paper we consider completely decomposable torsion-free groups and we determine the subgroups which are ideals in every ring over such groups.

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 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.

Let $R$ be a 2-torsion free prime ring and let $\sigma ,\tau$ be automorphisms of $R$. For any $x,y\in R$, set ${[x,y]}_{\sigma ,\tau }=x\sigma \left(y\right)-\tau \left(y\right)x$. Suppose that $d$ is a $(\sigma ,\tau )$-derivation defined on $R$. In the present paper it is shown that $\left(i\right)$ if $R$ satisfies ${[d\left(x\right),x]}_{\sigma ,\tau }=0$, then either $d=0$ or $R$ is commutative $\left(ii\right)$ if $I$ is a nonzero ideal of $R$ such that $\left[d\right(x),d(y\left)\right]=0$, for all $x,y\in I$, and $d$ commutes with both $\sigma$ and $\tau$, then either $d=0$ or $R$ is commutative. $\left(iii\right)$ if $I$ is a nonzero ideal of $R$ such that $d\left(xy\right)=d\left(yx\right)$, for all $x,y\in I$, and $d$ commutes with $\tau$, then $R$ is commutative. Finally a related result has been obtain for $(\sigma ,\tau )$-derivation....