On the Conjugacy Separability in the Class of Finite P -groups of Finitely Generated Nilpotent Groups

It is proved that for any prime p a finitely generated nilpotent group is conjugacy separable in the class of finite p-groups if and only if the tor-sion subgroup of it is a finite p-group and the quotient group by the torsion subgroup is abelian. 1. Let K be a class of groups. A group G is called residual K (or K-residual) if for each non-unit element a ∈ G there is a homomorphism ϕ of G onto some group X from the class K (or K-group) such that aϕ is not the identity. The group G is called conjugacy separable in the class K (or conjugacy K-separable), if whenever a and b are not conjugate in G, there is a homomorphism ϕ of G onto K-group X such that aϕ and bϕ are not conjugate in X. It is easy to see that a conjugacy K-separable group is also K-residual. Since in general the inverse statement is not true, it is interesting to find such classes of groups for which the property to be K-residual implies conjugacy K-separability. The most investigated (and chronologically first) is the case when class K is the class F of all finite groups; in this case one studies finite residuality and conjugacy separability respectively. K. Gruenberg [4] showed that finitely generated nilpotent groups are residually finite, and then N. Blackburn [5] proved that such groups are conjugacy separable. On the other hand the famous theorem of P. Hall states that every finitely generated metabelian group is residually finite, but there exists the example of finitely generated metabelian group that is not conjugacy separable constructed by M. I. Kargapolov and E. I. Timoshenko [1]. When a class of groups is proved to be F-residual or conjugacy F-separable, the question arises if these groups are K-residual or conjugacy K-separable for some subclass K of the class F. From this point of view the class F p of all finite p-groups is frequently considered. For example, A. I. Mal'cev [2] showed that free groups are residually finite. K. Gruenberg [4] proved that for any prime p every finitely generated torsion-free nilpotent group is F p-residual. From this assertion and from the theorem of W. Magnus about N-residuality of free groups (where N is the class of all finitely generated torsion-free nilpotent groups) follows that every free group is F p-residual for all primes p. Since [3] these …