Isomorphism of the Baumslag-Solitar groups

Necessary and sufficient conditions for the isomorphism of two groups, each defined by a single relation of the type a^–1b^ma=bⁿ, are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1684–1686, December, 1991.     

The intersection of the subgroups of finite index in Baumslag—Solitar groups

For any one-relator group in the family of Baumslag—Solitar groups, a system of its elements is indicated whose normal closure in the group coincides with the intersection of all normal finite-index subgroups. The well-known criterion for the residual finiteness of Baumslag—Solitar groups is an immediate consequence of this result. It is also shown that, if the intersection of all finite-index normal subgroups in a Baumslag—Solitar group differs from the identity subgroup (i.e., if the group is not residually finite), then this intersection cannot be the normal closure of any finite set of elements.     

On the residual properties of Baumslag–Solitar groups

A survey of results on the residual properties of Baumslag–Solitar groups that have been obtained to date. Some unpublished results are included and for certain results new proofs are given     

On the Isomorphism of Some Groups with One Defining Relation

Each of the groups in the class under consideration in the paper is presented by one defining relation depending on three integral parameters and is an HNN-extension of some Baumslag—Solitar group. Necessary and sufficient conditions for two groups of this class to be isomorphic and also for two nonisomorphic groups to be homomorphic images of each other are obtained. A consequence of this fact is the negative answer to Question 3.33 in “The Kourovka Notebook.”     

Residual finiteness of descending HNN-extension of groups

The paper examines the special case of the general construction of HNN-extensions of groups in which at least one of the associated subgroups is the base group. A criterion is determined for a group obtained in this way to be residually finite. Any group obtained as such an extension from a free nilpotent group of finite rank is residually finite.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 842–845, June, 1992.     

Residual Nilpotence of Groups with One Defining Relation

Mathematical Notes - All groups in the family of Baumslag-Solitar groups (i.e., groups of the form G(m,n) = 〈a, b; a?1bma = bn〉, where m and n are nonzero integers) for which...     

On a Family of Residually Finite Groups

Mathematical Notes - It is known that there exists a finitely generated residually finite group (for short, a residually F-group) the extension by which of some finite group is not a residually...