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In this paper we prove a theorem which extends a result due to H. Heineken. We prove that if $n(G)\simeq n(H)$ ($G$ hypercentral not locally cyclic p-group with property (P) in no. 1, $H^{\prime }$ hypercentral group) then $H$ is a hypercentral p-group. More generally: if $n(G)\simeq n(H)$ (G hypercentral torsion group, $H$ soluble group) then $H$ is a locally finite group.