A note on endomorphisms of hypercentral groups (Q1858183)

A subset \(X\) of a group \(G\) is called an `End-basis' of \(G\) if endomorphisms which agree on \(X\) are equal. There is no loss in assuming that \(X\) is a subgroup. If \(X\) is an End-basis of \(G\), then an endomorphism which is zero on \(X\) is zero. This is equivalent to \(\Hom(G/X^G,G)=0\): a subgroup with this property is called a `zero-basis' of \(G\). It is easy to show that for Abelian groups End-bases and zero-bases are the same thing. However the authors give an example of a locally nilpotent metabelian group which has a zero-basis that is not an End-basis. The main result of the paper is Theorem: Let \(H\) be a subnormal subgroup of a hypercentral group \(G\). If \(H\) is a zero-basis, then it is an End-basis of \(G\).