Fully inert subgroups of Abelian \(p\)-groups. (Q406603)

A subgroup \(H\) of an Abelian group \(G\) is \textit{fully inert} if the index \([\varphi(H):H\cap\varphi(H)]\) is finite for every endomorphism \(\varphi\) of \(G\). This paper is devoted to the study of fully inert subgroups of Abelian \(p\)-groups.NEWLINENEWLINE The main open problem considered is whether or not every fully inert subgroup of a given Abelian group \(G\) is commensurable with some fully invariant subgroup of \(G\). Recall that two subgroups \(K\) and \(L\) of \(G\) are \textit{commensurable} if \([K:L\cap K]\) and \([L:L\cap K]\) are both finite. The main result of this paper is a positive answer to the above question in the case when \(G\) is a direct sum of cyclic \(p\)-groups.NEWLINENEWLINE To prove this theorem, the authors study separately two cases. Firstly, they prove that for a bounded Abelian \(p\)-group \(G\), a fully inert subgroup of \(G\) is commensurable with some fully invariant subgroup of \(G\). Secondly, they show that the same conclusion holds true assuming \(G\) to be a semistandard Abelian \(p\)-group. As a final step, they combine the two results to get the general theorem.NEWLINENEWLINE Finally, using an Abelian \(p\)-group constructed by Pierce, an example is given of a separable Abelian \(p\)-group \(G\) containing some fully inert subgroup that is not commensurable with any fully invariant subgroup of \(G\).