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The paper develops algorithmic methods to enumerate all normal subgroups of a finitely presented group such that the factor groups are either isomorphic to PSL ( 2 , p n ) or to PGL ( 2 , p n ) . The case of two generators is treated in detail. A range of examples starting with the free group on two generators and ending with groups having only finitely many normal subgroups of this type is discussed.

V.N. Remeslennikov proposed in 1976 the following problem: is any countable abelian group a subgroup of the center of some finitely presented group ? We prove that every finitely generated recursively presented group G is embeddable in a finitely presented group K such that the center of G coincide with that of K . We prove also that there exists a finitely presented group H with soluble word problem such that every countable abelian group is embeddable in the center of H . This gives a strong positive answer to the question raised by V.N. Remeslennikov.

Graham Higman proved that the finitely generated groups that occur as subgroups of finitely presented groups are precisely those that can be defined by recursively enumerable sets of relations. We prove the analogue for lattice-ordered groups: Theorem A finitely generated lattice-ordered group is a sublattice subgroup of some finitely presented lattice-ordered group if and only if it can be defined by a recursively enumerable set of relations. Consequently, there is a universal finitely presented lattice-ordered group.

A finitely generated algebra A in a variety V is called finitely determined in V if there exists a finite V-consistent set of equalities and inequalities in an alphabet containing the generating set of A, which, together with the identities of V, yields all relations and non-relations of A. Obviously, if the equational theory of V is recursively enumerable then any finitely determined algebra in V has solvable word problem. The known algebraic characterizations of groups and semigroups with solvable word problem imply that in the varieties of all groups and all semigroups the members with solvable word problem are finitely determined. We construct a finitely generated center-by-metabelian group with solvable word problem, which is not finitely determined in every group variety V with ZA2⊆V⊆A3. We show that every extension of a finitely generated abelian group by a finite group from a variety W is finitely determined in every variety V⊇ZA2W. However, in any abelian-by-nilpotent variety no infinite group is finitely determined; moreover, in every variety, in which all finitely presented algebras are residually finite, each finitely determined algebra is finite. In the variety of all associative linear algebras over a finitely generated field every member with solvable word problem is finitely determined. We construct an example, which shows that for the variety of all associative rings it is not true; however, in this variety each torsion-free member with solvable word problem is finitely determined.

It is well known that the triviality problem for finitely presented groups is unsolvable; we ask the question of whether there exists a general procedure to produce a non-trivial element from a finite presentation of a non-trivial group. If not, then this would resolve an open problem by J. Wiegold: `Is every finitely generated perfect group the normal closure of one element?' We prove a weakened version of our question: there is no general procedure to pick a non-trivial generator from a finite presentation of a non-trivial group. We also show there is neither a general procedure to decompose a finite presentation of a non-trivial free product into two non-trivial finitely presented factors, nor one to construct an embedding from one finitely presented group into another in which it embeds. We apply our results to show that a construction by Stallings on splitting groups with more than one end can never be made algorithmic, nor can the process of splitting connect sums of non-simply connected closed 4-manifolds., Comment: 16 pages, revised version, minor corrections made, additional results on algorithms to construct embeddings and enumerate subgroups. v3: minor changes to typesetting and theorem numberings, minor corrections made