Let F be a field, A be a vector space over F, GL(F, A) be the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dimF(BFG/B) is finite. A subspace B is called almost G-invariant, if dim F(B/Core G(B)) is finite. In the current article, we study linear groups G such that every subspace of A is either nearly G-invariant or almost G-invariant in the case when G is a soluble p-group where p = char F.

Suppose $G$ is a $p$-mixed splitting abelian group and $R$ is a commutative unitary ring of zero characteristic such that the prime number $p$ satisfies $p\notin \text{inv}\left(R\right)\cup \text{zd}\left(R\right)$. Then $R\left(H\right)$ and $R\left(G\right)$ are canonically isomorphic $R$-group algebras for any group $H$ precisely when $H$ and $G$ are isomorphic groups. This statement strengthens results due to W. May published in J. Algebra (1976) and to W. Ullery published in Commun. Algebra (1986), Rocky Mt. J. Math. (1992) and Comment. Math. Univ. Carol. (1995).

∗ The work was supported by the National Fund “Scientific researches” and by the Ministry of Education and Science in Bulgaria under contract MM 70/91.Let K be a field of characteristic p > 0 and let G be a direct sum of cyclic groups, such that its torsion part is a p-group. If there exists a K-isomorphism KH ∼= KG for some group H, then it is shown that H ∼= G. Let G be a direct sum of cyclic groups, a divisible group or a simply presented torsion abelian group. Then KH ∼= KG as K-algebras...

Let K be a field of characteristic p > 2 and let G be a group. Necessary and sufficient conditions are obtained so that the group algebra KG is strongly Lie solvable of derived length at most 3. It is also shown that these conditions are equivalent to KG Lie solvable of derived length 3 in characteristic p ≥ 7.

Let A be an R G-module, where R is an integral domain and G is a soluble group. Suppose that C G(A) = 1 and A/C A(G) is not a noetherian R-module. Let L nnd(G) be the family of all subgroups H of G such that A/C A(H) is not a noetherian R-module. In this paper we study the structure of those G for which L nnd(G) satisfies the maximal condition.

Let $G$ be an abelian group, $R$ a commutative ring of prime characteristic $p$ with identity and $R_{t}G$ a commutative twisted group ring of $G$ over $R$. Suppose $p$ is a fixed prime, $G_p$ and $S\left(R_{t}G\right)$ are the $p$-components of $G$ and of the unit group $U\left(R_{t}G\right)$ of $R_{t}G$, respectively. Let $R^{*}$ be the multiplicative group of $R$ and let $f_{\alpha }\left(S\right)$ be the $\alpha$-th Ulm-Kaplansky invariant of $S\left(R_{t}G\right)$ where $\alpha$ is any ordinal. In the paper the invariants $f_n\left(S\right)$, $n\in \mathbb{N}\cup \left\{0\right\}$, are calculated, provided $G_p=1$. Further, a commutative ring $R$ with identity of prime characteristic $p$ is said...

Let $R$ be a commutative ring, $M$ an $R$-module and $G$ a group of $R$-automorphisms of $M$, usually with some sort of rank restriction on $G$. We study the transfer of hypotheses between $M/C_M\left(G\right)$ and $[M,G]$ such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose $[M,G]$ is $R$-Noetherian. If $G$ has finite rank, then $M/C_M\left(G\right)$ also is $R$-Noetherian. Further, if $[M,G]$ is $R$-Noetherian and if only certain abelian sections...

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.