Profinite 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 ) are any automorphisms of S, and qj (j = 1, . . . , M ) are...

36p noel_noel 17-01-2013 20 5   Download

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let w be a ‘locally finite’ group word and d ∈ N. Then there exists f = f (w, d) such that in every d-generator finite group G, every element of the verbal subgroup w(G) is equal to a product of f w-values.

69p noel_noel 17-01-2013 18 5   Download

In 1970 Alexander Grothendieck [6] posed the following problem: let Γ1 and Γ2 be finitely presented, residually finite groups, and let u : Γ1 → Γ2 be a ˆ homomorphism such that the induced map of profinite completions u : Γ1 → Γ2 ˆ ˆ is an isomorphism; does it follow that u is an isomorphism? In this paper we settle this problem by exhibiting pairs of groups u : P → Γ such that Γ is a direct product of two residually finite, hyperbolic groups, P is a finitely presented subgroup of infinite index, P is not...

16p tuanloccuoi 04-01-2013 26 5   Download