Previous |  Up |  Next


vector space; linear groups; periodic groups; soluble groups; invariant subspaces
Let $F$ be a field, $A$ be a vector space over $F$, $\operatorname{GL}(F,A)$ be the group of all automorphisms of the vector space $A$. A subspace $B$ is called almost $G$-invariant, if $\dim _{F}(B/\operatorname{Core}_{G}(B))$ is finite. In the current article, we begin the study of those subgroups $G$ of $\operatorname{GL}(F,A)$ 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.
[BLNSW] Buckley J.T., Lennox J.C., Neumann B.H., Smith H., Wiegold J.: Groups with all subgroups normal-by-finite. J. Austral. Math. Soc. Ser. A 59 (1995), 384–398. DOI 10.1017/S1446788700037289 | MR 1355229 | Zbl 0853.20023
[DK] Drozd Yu.A., Kirichenko V.V.: Finite Dimensional Algebras. Vyshcha shkola, Kyiv, 1980. MR 0591671 | Zbl 0816.16001
[FL] Fuchs L.: Infinite Abelian Groups. Vol. 1. Academic Press, New York, 1970. MR 0255673 | Zbl 0338.20063
[KM] Kargapolov M.I., Merzlyakov Yu.I.: The Foundations of Group Theory. Nauka, Moscow, 1982. MR 0677282
[KW] Kegel O.H., Wehrfritz B.A.F.: Locally Finite Groups. North-Holland, Amsterdam, 1973. MR 0470081 | Zbl 0259.20001
[KOS] Kurdachenko L., Otal J., Subbotin I.: Artinian Modules Over Group Rings. Frontiers in Mathematics, Birkhäuser, Basel, 2007. MR 2270897 | Zbl 1110.16001
[KSaSu] Kurdachenko L.A., Sadovnichenko A.V., Subbotin I.Ya.: On some infinite dimensional groups. Cent. Eur. J. Math. 7 (2009), no. 2, 176–185. DOI 10.2478/s11533-009-0007-6 | MR 2506958
[PR] Pierce R.S.: Associative Algebras. Springer, Berlin, 1982. MR 0674652 | Zbl 0671.16001
[RD] Robinson D.J.S.: Finiteness Conditions and Generalized Soluble Groups. Part 1, Springer, New York, 1972. MR 0332989 | Zbl 0243.20033
[WB] Wehrfritz B.A.F.: Infinite Linear Groups. Springer, Berlin, 1973. MR 0335656 | Zbl 0261.20038
Partner of
EuDML logo