Fully inert subgroups of free Abelian groups. (Q484560)

A subgroup \(H\) of an Abelian group \(G\) is fully inert if for every \(\varphi\in\text{End}(G)\), \((\varphi H+H)/H\) is finite. The notion is used to study the dynamical properties of endomorphisms. In this paper, the authors continue their previous research [J. Group Theory 16, No. 6, 915-939 (2013; Zbl 1292.20062)] on the topic by characterising the fully inert subgroups of a free group, \(G\), in the non-trivial case that \(G\) has infinite rank. The main result states that \(H\) is fully inert in \(G\) if and only if (1) \(G/H\) is bounded; (2) for each prime \(p\), at most one Ulm-Kaplansky invariant of the \(p\)-component \(M_p\) of \(G/H\) is infinite, and equal to the rank of \(G\); (3) The rank of the elementary \(p\)-group \((H+pG)/pG\) is finite for each prime \(p\) such that \(M_p\) is infinite. The proof consists of an ingenious use of a sequence of elementary lemmas.