Given a finite group G, let ram(G) (resp. ramt(G)) denote the minimal positive integer such that G can be realized as the Galois group of an extension of Q (resp. a tamely ramified extension of Q) ramified only at ram(G) (resp. ramt(G)) finite primes. The present paper is devoted to study ram(G) and ramt(G), for some solvable groups G. More precisely, we consider the case where G is either a fi...

For every finitely generated recursively presented group G we construct a finitely presented group H containing G such that G is (Frattini) embedded into H and the group H has solvable conjugacy problem if and only if G has solvable conjugacy problem. Moreover G and H have the same r.e. Turing degrees of the conjugacy problem. This solves a problem by D. Collins.