Morphic \(p\)-groups. (Q392125)

A group \(G\) is said to be `morphic' if every morphism \(\alpha\colon G\to G\) for which \(\text{im}(\alpha)\) is \(G\)-invariant satisfies \(G/\text{im}(\alpha)\cong\ker(\alpha)\). It appears that morphic \(p\)-groups of orders \(p^4\), \(p^5\) and \(p^6\) are Abelian. If \(G\) is a morphic \(p\)-group such that either \(d(G)=2\) or \(|G'|<p^3\), then it is Abelian. Necessary and sufficient conditions for a morphic \(p\)-group to be Abelian are given.