Groups with an element permuting with finitely many of its conjugates

We study the problem as to which is the cardinality of connected components of the graph Γ_α, defined as follows. Let G be a group and a an element of G. The vertex set V(Γ_α) of the graph is the conjugacy class of elements,Cl _G(a), and two vertices x and y of the graph Γ_α are bridged by an edge iff x=y. If the intersectionC _G(a)∩Cl _G(a) is finite, Γ_α is locally finite. We prove that connected components of the locally finite graph Γ_α are finite in some classes of groups. Supported by RFFR grant No. 94-01-01084. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 543–551, September–October, 1996.     

Groups containing an element that permutes with a finite number of its conjugates

We study a group G containing an element g such that C_G(g)∩g^G is finite. The nonoriented graph Γ is defined as follows. The vertex set of Γ is the conjugacy class g^G. Vertices x and y of the graph G are bridged by an edge iff x≠y and xy=yx. Let Γ₀ be some connected component of G. On a condition that any two vertices of Γ₀ generate a nilpotent group, it is proved that a subgroup generated by the vertex set of Γ₀ is locally nilpotent. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 637–650, November–December, 1998.     

Metrics with only finitely many isometry invariant geodesics

Dedicated to Professor Wilhelm Klingenberg with best wishes on his 65 th birthday     

Irreducible representations of finitely generated nilpotent torsion-free groups

An investigation of the structure of the quotient algebra, with respect to a prime ideal, of the group algebra of a finitely-generated nilpotent torsion-free group. Conditions are studied under which an irreducible representation of such a group is induced.Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 199–210, February, 1971.     

On monomial representations of finitely generated nilpotent groups

A result of Segal states that every complex irreducible representation of a finitely generated nilpotent group G is monomial if and only if G is abelian-by-finite. A conjecture of Parshin, recently proved affirmatively by Beloshapka and Gorchinskii (2016), characterizes the monomial irreducible representations of finitely generated nilpotent groups. This article gives a slightly shorter proof of the conjecture using ideas of Kutzko and Brown. We also give a characterization of the finite-dimensional irreducible representations of two-step nilpotent groups and describe these completely for two-step groups whose center has rank one.     

Locally nilpotent groups with a centralizer of finite rank

Locally nilpotent groups in which the centralizer of some finitely generated subgroup has finite rank are studied. It is shown that if G is such a group and F is a finitely generated subgroup with centralizer C_G(F) of finite rank, then the centralizer of the image of F in the factor group G/t(G) modulo the periodic part t(G) also has finite rank. It is also shown that G is hypercentral when F is cyclic and either G is torsion-free or all Sylow subgroups of the periodic part of C_G(F) are finite.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1511–1517, November, 1992.     

Decompositions of finitely generated nilpotent groups of class 2

In this paper we prove that rational indecomposability is a genus property for finitely generated torsion-free nilpotent groups of class 2. We use this result to determine the genus of finitely generated torsion-free nilpotent groups of class 2 which decompose as a direct product of rationally indecomposable groups. Received: 3 November 2005