On the relationship between the localization of groups and that of relative groups (Q1814411)

Let \(G\) be a group, \(P\) a set of prime numbers and \(G_ P\) be the localization of \(G\) as defined by \textit{P. Ribenboim} [Lect. Notes Math. 740, 444-456 (1979; Zbl 0425.20032)]. For a normal subgroup \(N\) of \(G\), let \((N_{(P)},G_{(P)})\) be the localization of the pair \((N,G)\) in the sense of \textit{P. Hilton} [Math. Z. 132, 263-286 (1973; Zbl 0264.20037)]. The main result of the paper claims that the groups \(G_ P\) and \(G_{(P)}\) are isomorphic provided the quotient group \(G/N\) is \(P\)- local periodic. \{The commutative diagram as in the statement of Theorem 1 is left out from printing/typing on page 126\}.