On the socles of fully inert subgroups of abelian \(p\)-groups (Q2029719)

The authors define the fully inert socle-regular and weakly fully inert socle-regular abelian \(p\)-groups. This notions are generalizations of the notion socle-regularity abelian \(p\)-group, which was introduced by \textit{P. Danchev} et al. [Arch. Math. 92, No. 3, 191--199 (2009; Zbl 1173.20038)]. An abelian \(p\)-group \(G\) is said to be fully inert socle-regular if, for all infinite fully inert subgroups \(H\) of \(G\), there exists an ordinal \(\alpha\), depending on \(H\), such that \(H[p] \sim p^{\alpha}G[p]\) and \(G\) is said to be weakly fully inert socle-regular if, for all infinite fully inert subgroups \(H\) of \(G\), there exists an ordinal \(\alpha\), depending on \(H\), such that \(p^{\alpha}G \neq 0\) and \(H[p] \cap p^{\alpha}G[p]\) is of finite index in \(p^{\alpha}G[p]\). It is proved that in the case of groups of length \(\omega\), these two group classes coincide, but that in the case of groups of length \(\omega +1\), they differ. Some results about direct sums and direct products these groups are obtained. It is proved that if \(G\) is a group with a finite index subgroup \(A\), then \(G\) is fully inert socle-regular if, and only if, \(A\) is fully inert socle-regular. Some open problems are formulated.