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We show that for every prime $r$ all $r$-subgroups in the normalized units ofthe integral group ring of $\operatorname{PSL}(2,p^3)$ are isomorphic tosubgroups of $\operatorname{PSL}(2,p^3)$. This answers a question of M.Hertweck, C.R. H\"ofert and W. Kimmerle for this series of groups.

Torsion subgroups in the units of the integral group ring of PSL(2,p^3)