SOME RESIDUAL PROPERTIES OF SOLUBLE GROUPS OF FINITE RANK

The generalization of one classical Smel’kin’s theorem for polycyclic groups is obtained. A. L. Smelkin proved that if G is a polycyclic group, then it is a virtually residually finite p-group for any prime p. Recall that a group G is said to be a residually finite p-group if for every nonidentity element a of G there exists a homomorphism of the group G onto some finite p-group such that the image of the element a differs from 1. A group G will be said to be a virtually residually finite p-group if it contains a finite index subgroup which is a residually finite p-group. One of the generalizations of the notation of polycyclic group is a notation of soluble finite rank group. Recall that a group G is said to be a group of finite rank if there exists a positive integer r such that every finitely generated subgroup in G is generated by at most r elements. For soluble groups of finite rank the following necessary and sufficient condition to be a residually finite π-group for some finite set π of primes is obtained. If G is a group of finite rank, then the group G is a residually finite π- group for some finite set π of primes if and only if G is a reduced poly-(cyclic, quasicyclic, or rational) group. Recall that a group G is said to be a reduced group if it has no nonidentity radicable subgroups. A group H is said to be a radicable group if every element h in H is an mth power of an element of H for every positive number m. It is proved that if a soluble group of finite rank is a residually finite π- group for some finite set π of primes, then it is a virtually residually finite nilpotent π-group. We prove also the following generalization of Smel’kin’s theorem. Let π be a finite set of primes. If G is a soluble group of finite rank, then the group G is a virtually residually finite π-group if and only if G is a reduced poly-(cyclic, quasicyclic, or rational) group and G has no π-radicable elements of infinite order. Recall that an element g in G is said to be π-radicable if g is an mth power of an element of G for every positive π-number m.

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