Approximability of finite rank soluble groups by certain classes of finite groups
A group \(G\) is said to be \textit{residually finite} (respectively, a \textit{residually finite \(\pi\)-group} for some finite set \(\pi\) of primes) if the intersection of all normal subgroups \(N\) of \(G\) such that \(G/N\) is fnite (respectively, is a fnite \(\pi\)-group) is trivial. In [J. Lond. Math. Soc. 27, 81--85 (1952; Zbl 0046.02003)], \textit{K. A. Hirsch} proved that ``every polycyclic group is residually fnite''. This result was improved by \textit{A. Learner} [Ill. J. Math. 8, 536--542 (1964; Zbl 0124.26304)], who proved that a polycyclic group is a residually finite \(\pi\)-group for some finite set \(\pi\). Later, \textit{A. L. Shmel'kin} [Sib. Mat. Zh. 9, 234--235 (1968; Zbl 0203.32602)] showed that a polycyclic group is a residually finite \(p\)-group for every prime \(p\). In the article under review, the author proves, among other things, the following results: Theorem 1. Let \(G\) be a soluble group with finite rank, and let \(T(G)\) be the maximal normal periodic subgroup of \(G\). 1. The group \(G\) is a residually finite \(\pi\)-group, for some finite prime numbers set \(\pi\), iff \(T(G)\) is finite and \(G\) does not contain subgroups isomorphic to the additive rational number group \(Q\). 2. Let \(p\) be a prime number. Then, \(G\) is a finite extension of a residually finite \(p\)-group iff \(T(G)\) is finite and G does not contain subgroups isomorphic to the additive group \(Q_p\) of \(p\)-adic fractions. Theorem 2. Let \(G\) be a nilpotent group with finite rank, and let \(T(G)\) be the torsion subgroup of \(G\). Then, \(G\) is a residually finite \(p\)-group for some prime \(p\), iff \(T(G)\) is a finite \(p\)-group and \(G\) does not contain subgroups isomorphic to the additive group \(Q_p\) of \(p\)-adic fractions.
scientific article; zbMATH DE number 6362829
scientific article