Projectively invariant subgroups of abelian \(p\)-groups (Q2037749)

All groups are abelian. A subgroup \(C\) of a group \(G\) is said to be projectively invariant (briefly, a pi-subgroup) if \(\pi(C) \leq C\) for every projection \(\pi\) of \(G\). These groups have been studied by several authors, for example by \textit{C. K. Megibben} [Tamkang J. Math. 8, 177--182 (1977; Zbl 0421.20023)], \textit{J. Hausen} [Tamkang J. Math. 12, 215--218 (1981; Zbl 0493.16022)] and by author of the paper under this review [J. Math. Sci., New York 164, No. 1, 143--147 (2010; Zbl 1288.20076); translation from Fundam. Prikl. Mat. 14, No. 6, 211--218 (2008)]. Let \(C\) be a subgroup of a reduced \(p\)-group \(G\). With the subgroup \(C\), the author associates the sequence \(H(C)=(\alpha_0(C),\alpha_1(C),\ldots)\) where \(\alpha_k(C)=\min h_G(p^kC[p])\). The main result of the paper is the next theorem: if \(C\) is a projectively invariant subgroup of a reduced \(p\)-group \(G\) and \(\alpha_k(C)\) is integer number for any \(k\in \mathbb{N_0}\) then the subgroup \(C\) is fully invariant.