On supersolubility in some groups with finitely generated Fitting radical

On manifolds with finitely generated homotopy groups

Let G be an infinite group which is finitely presented. Let X be a finite CW−complex of dimension q whose fundamental group is Z × G. We prove that for some i ≤ q the homotopy group πi(X) is not finitely generated. Let M be a manifold of dimension n whose fundamental group is Zn−2×G. Then the same conclusion holds (for some i ≤ maxn2 ] , 3 } ) unless M is an Eilenberg-McLane space. In particula...

Finitely Generated Abelian Groups

Finitely Generated Semiautomatic Groups

The present work shows that Cayley automatic groups are semiautomatic and exhibits some further constructions of semiautomatic groups and in particular shows that every finitely generated group of nilpotency class 3 is semiautomatic.

Powers in Finitely Generated Groups

In this paper we study the set Γn of nth-powers in certain finitely generated groups Γ. We show that, if Γ is soluble or linear, and Γn contains a finite index subgroup, then Γ is nilpotent-by-finite. We also show that, if Γ is linear and Γn has finite index (i.e. Γ may be covered by finitely many translations of Γn), then Γ is soluble-by-finite. The proof applies invariant measures on amenable...

On finitely generated profinite groups, II: products in quasisimple groups

We prove two results. (1) There is an absolute constant D such that for any finite quasisimple group S, given 2D arbitrary automorphisms of S, every element of S is equal to a product of D ‘twisted commutators’ defined by the given automorphisms. (2) Given a natural number q, there exist C = C(q) and M = M(q) such that: if S is a finite quasisimple group with |S/Z(S)| > C, βj (j = 1, . . . ,M) ...