Prosolvability criteria and properties of the prosolvable radical via Sylow sequences. (Q285585)

The authors generalize two solvability criteria for finite groups to the profinite case, proving two prosolvability criteria. The first criterion characterises prosolvability by the non-existence of Thompson triples (also called \(T\)-triples), that is, triples \((a,b,c)\) of non-identity elements of pairwise coprime order (in the profinite sense) such that \(abc=1\). The second criterion characterises prosolvability by the possibility of factorizing \(G\) as a product of Sylow subgroups associated to distinct primes in a given (or arbitrary) order (giving a suitable definition of what this means in the profinite case). Also, they characterise the prosolvable residual \(S(G)\) as the subgroup generated by the Thompson factors (i.e. the elements belonging to some Thompson triple) and they deal with the open problem of showing that the prosolvable radical \(R(G)\) equals the intersection of all the Sylow products (where a Sylow product is a suitable profinite version of a product of Sylow subgroups associated each to a different prime number involved). They reduce this problem to the finite group case.