Let H be a subgroup of a group G. H is said to be S-embedded in G if G has a normal T such that HT is an S-permutable subgroup of G and H ∩ T ≤ H sG, where H denotes the subgroup generated by all those subgroups of H which are S-permutable in G. In this paper, we investigate the influence of minimal S-embedded subgroups on the structure of finite groups. We determine the structure the finite grou...

This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. Let L denote the fundamental group of the link consisting of a chain of 4 circles. It is shown that L is not subgroup separable. Furthermore, it is sh...

Following Bass–Milnor–Serre, we prove that SLn(Z) has the congruence subgroup property for n ≥ 3. This was originally proved by Mennicke and Bass–Lazard–Serre. Let Γn = SLn(Z). The congruence subgroup problem for Γn (solved independently by Mennicke [Me] and Bass–Lazard–Serre [BLS]) seeks to classify all finite-index subgroups of Γn. For l ≥ 2, the level l principal congruence subgroup of Γn, d...

Homework Policy. Please read through all the problems. I will be happy to discuss the solutions during office hours. Problems. Problem 1. For a finite type, separated k-scheme X, recall the alternative definition of the subgroup Ratl(X) ⊂ Zl(X). It equals the subgroup generated by all pushforward classes, i∗[W0]− i∗[W∞], for all k-schemes W of pure dimension l + 1, and for all proper morphisms ...

We shall term a group G supersoluble if every homomorphic image H9*l of G contains a cyclic normal subgroup different from 1. Supersoluble groups with maximum condition, in particular finite supersoluble groups, have been investigated by various authors: Hirsch, Ore, Zappa and more recently Huppert and Wielandt. In the present note we want to establish the close connection between supersoluble ...

We study a measure entropy for finitely generated free group actions called f-invariant entropy. The f-invariant entropy was developed by L. Bowen and is essentially a special case of his measure entropy theory for actions of sofic groups. In this paper we relate the f-invariant entropy of a finitely generated free group action to the f-invariant entropy of the restricted action of a subgroup. ...

We are concerned with torsion subgroups of Lie groups. We extend the classical result of C. Jordan on the structure of finite linear groups to torsion subgroups of connected Lie groups. 1. In [1], Boothby and Wang proved that, for any connected Lie group G, there exists a number k{G) such that any finite subgroup contains an abelian normal subgroup whose index is bounded by k{G), thereby genera...