On commutator socle-regular Abelian \(p\)-groups. (Q741274)

A subgroup \(H\) of an Abelian \(p\)-group \(G\) is called commutator invariant if for all endomorphisms \(\varphi\) and \(\psi\) of \(G\), \((\phi\psi-\psi\phi)(H)\subseteq H\), and projection invariant if for all idempotent endomorphisms \(\pi\), \(\pi(H)\subseteq H\). The authors show that these are independent generalizations of fully invariant, and find various conditions on \(G\) for which they coincide. They define \(G\) to be commutator socle-regular (cs-r) if for each commutator invariant subgroup \(H\), \(H[p]=p^\alpha G[p]\) for some ordinal \(\alpha\). They show that this property is determined by the reduced part of \(G\) and is inherited by large subgroups of \(G\). They describe several necessary or sufficient conditions for a group \(G\) to be cs-r. In the last section of the paper, the authors generalize the notion of cs-r to several properties of the form \(G\) is \(P\)-socle regular if subgroups \(H\) in the class \(P\) satisfy \(H[p]=p^\alpha G[p]\), where \(P\) is the class of fully invariant, characteristic or projection invariant subgroups. They investigate the relations among these properties and consider classes of groups for which they coincide. In general, the proofs consist of extensive commutator calculations.