Free subgroups of groups with a single defining relation

Summary This paper proves the following theorem. LetG be a group generated byA andB. LetW be a word inA andB which in cyclically reduced form contains bothA andB. IfG has the presentationG= ^u =I>, u =2, then there exists an integerv ₀ such thatA, B ^v generate a free subgroup ofG for allv =v ₀. A suitable value forv ₀ is easily calculated. The result can be generalized by replacingW ^u byWW whereW is obtained fromW by means of applying certain automorphisms of period 2 toW. Here it is required that the wordWW satisfies an easily verifiable extra condition.     

Metabelian groups with a single defining relation and the Magnus embedding

Translated from Matematicheskie Zametki, Vol. 47, No. 4, pp. 597–605, April, 1995.     

Nonassociative rings with a single defining relation whose elementary theories are decidable

Translated from Algebra i Logika, Vol. 29, No. 5, pp. 509–522, September–October, 1990.     

Residual finiteness of a class of semigroups presented by a single relation

Communicated by Dominique Perrin     

A Lie algebra with a single defining relation need not be residually finite

Translated from Matematicheskie Zametki, Vol. 51, No. 4, pp. 3–7, April, 1992.     

On pro-p-groups with a single defining relator

We find necessary and sufficient conditions for the factor groups of the derived series of a pro-p-group with a single defining relation to be torsion free. For such groupsG we prove that the group algebra ℤ _pG is a domain and the cohomological dimension ofG is at most 2.     

On pro-p-groups with a single defining relator

Inventiones mathematicae -     

Transformation semigroups preserving a binary relation

We investigate the semigroups of full and partial transformations of a set X which preserve a binary relation σ defined on X. We consider in detail the case where σ is an order or a quasi-order relation. There are conditions of regularity of such semigroups. We introduce two definitions of preservation of σ for the semigroup of binary relations. It is proved that subsets of B(X) preserving σ are semigroups in each case. We give the condition of regularity of B _σ (X) in the case where σ(X) is a quasi-order.