Residual properties of abelian groups

Let п be a set of primes. For Abelian groups, the necessary and sufficient condition to be a virtually residually finite п-group is obtained, as well as a characterization of potent Abelian groups. Recall that a group G is said to be a residually finite п-group if for every nonidentity element a of G there exists a homomorphism of the group G onto some finite п-group such that the image of the element a differs from 1. A group G is said to be a virtually residually finite п-group if it contains a finite index subgroup which is a residually finite п-group. Recall that an element g in G is said to be п-radicable if g is an mth power of an element of G for every positive п-number m. Let A be an Abelian group. It is well known that A is a residually finite п-group if and only if A has no nonidentity п-radicable elements. Suppose now that п does not coincide with the set П of all primes. Let п' be the complement of п in the set П. And let T be a п'-component of A, i.e., T be a set of all elements of A whose orders are finite п'-numbers. We prove that the following three statements are equivalent to each other: (1) the group A is a virtually residually finite п-group; (2) the subgroup T is finite and the quotient group A / T is a residually finite п-group; (3) the subgroup T is finite and T coincides with the set of all п-radicable elements of A.

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