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.