Some isolated subsets of infinite solvable groups

Abstract

The main theorem of this paper offers necessary and sufficient conditions for a solvable group G to be covered by a finite union of certain types of isolated subsets.This result will have applications to the study of the semisimplicity problem for group rings of solvable groups.Let H be a subgroup of G.We definefor some m ^ 1} .Observe that i/ίΓ need not be a subgroup of G even if G is solvable.We say that H has locally finite index in G and write [G: H] = l.f.if for every finitely generated subgroup L of G we have [L: L Π H] < °o.Suppose [G: H] = l.f. and let x e G. Then [(x): {x) Π H] < oo so α; m e H for some m ^ 1 and x e i/I?.Thus G = VΊR.The main result of this paper is a generalized converse of this fact for solvable groups G. THEOREM.Let G be a solvable group and let H ly H 2 , •••, H n be subgroups with G =

Locations

• Pacific Journal of Mathematics - View - PDF
• Project Euclid (Cornell University) - View - PDF