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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 $B$ is called almost $G$ -invariant, if ${dim}_{F} (B / {Core}_{G} (B))$ is finite. In the current article, we begin the study of those subgroups $G$ of $GL (F, A)$ for which every subspace of $A$ is almost $G$ -invariant. More precisely, we consider the case when $G$ is a periodic group. We prove that in this case $A$ includes a $G$ -invariant subspace $B$ of finite codimension whose subspaces are $G$ -invariant.

Infinite dimensional linear groups with many G - invariant subspaces

Leonid Kurdachenko, Alexey Sadovnichenko, Igor Subbotin (2010)

Open Mathematics

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.

Infinite groups with many permutable subgroups.

A. Ballester-Bolinches, L. A. Kurdachenko, J. Otal, Pedraza T. (2008)

Revista Matemática Iberoamericana

Infinite locally finite groups of type PSL(2,K) or Sz(K) are not minimal under certain conditions.

Javier Otal, Juan Manuel Peña (1988)

Publicacions Matemàtiques

In classifying certain infinite groups under minimal conditions it is needed to find non-simplicity criteria for the groups under consideration. We obtain some of such criteria as a consequence of the main result of the paper and the classification of finite simple groups.

Infinite locally soluble $k$ -Engel groups

Lucia Serena Spiezia (1992)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this paper we deal with the class $E_{k}^{*}$ of groups $G$ for which whenever we choose two infinite subsets $X$ , $Y$ there exist two elements $x \in X$ , $y \in Y$ such that $[x, \underset{k}{\underset{︸}{y, \dots, y}}] = 1$ . We prove that an infinite finitely generated soluble group in the class $E_{k}^{*}$ is in the class $E_{k}$ of $k$ -Engel groups. Furthermore, with $k = 2$ , we show that if $G \in E_{2}^{*}$ is infinite locally soluble or hyperabelian group then $G \in E_{2}$ .

Infinite Soluble Groups with Engel Cycles; a Finiteness Condition.

Rolf Brandl (1983)

Mathematische Zeitschrift

Infinite soluble groups with no outer automorphisms

Derek J. S. Robinson (1980)

Rendiconti del Seminario Matematico della Università di Padova

Infinite Soluble Groups with the Bell Property: A Finiteness Condition.

Rolf Brandl (1987)

Monatshefte für Mathematik

Infinite two-generator groups of class two and their non-abelian tensor squares.

Sarmin, Nor Haniza (2002)

International Journal of Mathematics and Mathematical Sciences

Infinitely Generated Subgroups of Finitely Presented Groups. Part II.

Gilbert, Cannonito, F.B. Baumslag (1980)

Mathematische Zeitschrift

Infinitely Generated Subgroups of Finitely Presented Groups.I.

Gilbert Baumslag, Frank B. Cannonito (1977)

Mathematische Zeitschrift

Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant

Gwénaël Massuyeau (2012)

Bulletin de la Société Mathématique de France

Let $Σ$ be a compact connected oriented surface with one boundary component, and let $π$ be the fundamental group of $Σ$ . The Johnson filtration is a decreasing sequence of subgroups of the Torelli group of $Σ$ , whose $k$ -th term consists of the self-homeomorphisms of $Σ$ that act trivially at the level of the $k$ -th nilpotent quotient of $π$ . Morita defined a homomorphism from the $k$ -th term of the Johnson filtration to the third homology group of the $k$ -th nilpotent quotient of $π$ . In this paper, we replace groups...

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