Fitting subgroups and profinite completions of polycyclic groups

Profinite groups, profinite completions and a conjecture of Moore

Let R be any ring (with 1), Γ a group and RΓ the corresponding group ring. Let H be a subgroup of Γ of finite index. Let M be an RΓ−module, whose restriction to RH is projective. Moore’s conjecture [5]: Assume for every nontrivial element x in Γ, at least one of the following two conditions holds: M1) 〈x〉 ∩ H 6= {e} (in particular this holds if Γ is torsion free) M2) ord(x) is finite and invert...

Decision Problems and Profinite Completions of Groups

We consider pairs of finitely presented, residually finite groups P ↪→ Γ for which the induced map of profinite completions P̂ → Γ̂ is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not P is isomorphic to Γ. We construct pairs for which the conjugacy problem in Γ can be solved in quadratic time but the conjugacy problem in P is uns...

Maximal Subgroups in Finite and Profinite Groups

We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every finite group F is obtained as a quotient of an open subgroup of G. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann....

Counting the Closed Subgroups of Profinite Groups

The sets of closed and closed-normal subgroups of a profinite group carry a natural profinite topology. Through a combination of algebraic and topological methods the size of these subgroup spaces is calculated, and the spaces partially classified up to homeomorphism.

Random Normal Subgroups of Free Profinite Groups