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The unsolvability of the problem of deciding whether a class of finitely presented groups in a (p+3)-letter alphabet has Markov group properties is proved (p is the number of generators of the finitely presented group having a particular property of the kind in question). The problem of deciding whether a class of finitely presented groups in the minimal (p+l)-letter alphabet has Markov properties such that a group having those properties contains an infinite cyclic subgroup is proved to be unsolvable.

The main theorem of this paper states that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. It follows that every countable A belian group, and every countable locally finite group can be so embedded; and that there exists a finitely presented group which simultaneously embeds all finitely presented groups. A nother corollary of the theorem is the known fact that there exist finitely presented groups with recursively insoluble word problem . A by-product of the proof is a genetic characterization of the recursively enumerable subsets of a suitable effectively enumerable set.

Publisher Summary C. D. Papakyriakopoulos has recently proved Dehn's lemma. His proof has the merit that the basic construction, the tower, and the crucial lemmas apply to the sphere theorem as well as Dehn's lemma. However, if one is content with Dehn's lemma, the proof can be simplified. This chapter presents a simplified proof of Dehn's lemma and also presents a proof of an analogous theorem for surfaces with more than one boundary curve. By surface of type (p,r) it is meant that a connected, compact, orientable surface of genus p with a boundary consists of r 1-spheres. Thus, the Euler characteristic of such a surface is 2 (1–p)–r.

Publisher Summary Boone and Higman have recently proved that a finitely generated group has a solvable word problem if and only if (iff) it can be embedded in a simple group, which, is in turn, embeddable in a finitely presented group. The Boone–Higman theorem is a variation and mild extension of the universal algebraic result that a finitely presented simple algebra has a solvable word problem. This chapter presents a general framework for particular results. The purpose of the chapter is to give the universal algebraic version of the first half of the Boone–Higman result and give, for some particular algebra, the analogue of the second half up to but not including the embedding in a finitely presented algebra.