A subgroup \(H\) of a group \(G\) is said to be \textit{quasinormal} if \(HX=XH\) for all subgroups \(X\) of \(G\). A well-known theorem by \textit{R. Maier} and \textit{P. Schmid} [Math. Z. 131, 269-272 (1973; Zbl 0259.20017)] shows that every core-free quasinormal subgroup of a finite group \(G\) lies in the hypercentre of \(G\). This result fails to be true for an arbitrary infinite group, even in the metabelian case. However, a number of possible generalizations of Maier-Schmid theorem have been obtained, and in particular \textit{G. Busetto} [Rend. Semin. Mat. Univ. Padova 63, 269-284 (1980; Zbl 1319.20028)] proved that any periodic locally cyclic core-free quasinormal subgroup of an arbitrary group is contained in the hypercentre. In the paper under review, the authors show that a corresponding result holds for torsion-free locally cyclic quasinormal subgroups.