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.

Let G be a finite group which is not a cyclic p-group, p a prime number. We define an undirected simple graph ∆(G) whose vertices are the proper subgroups of G, which are not contained in the Frattini subgroup of G and two vertices H and K are joined by an edge if and only if G = ⟨H,K⟩. In this paper we classify finite groups with planar graph. Our result shows that only few groups have planar ...