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.

Algebraic properties of lattices of quotients of nite posets are considered. Using the known duality between the category of all nite posets together with all order-preserving maps and the category of all nite distributive (0; 1)-lattices together with all (0; 1)-lattice ho-momorphisms, algebraic and arithmetic properties of maximal proper sublattices and, in particular, Frattini sublattices of...

In this paper we prove that if G is a group for which there are k non-Frattini chief factors isomorphic to characteristically simple A, then has normal section C/R the direct product of minimal subgroups G/R A. This significant extension notion crown factors.

In this note we show that if two critical algebras generate the same congruence-permutable variety, then the varieties generated by their proper sections also coincide.