From the introduction: If \((G,\cdot)\) is a group, let \((M(G),+,\circ)\) be the set of all mappings from \(G\) to \(G\), with pointwise addition and with the composition of mappings as multiplication: if \(g\in G\) and \(\alpha,\beta\in M(G)\), by definition \(g(\alpha+\beta)=(g\alpha)\cdot(g\beta)\) and \(g(\alpha\circ\beta)=(g\alpha)\beta\). Then \((M(G),+,\circ)\) is a near-ring.
Let \((\text{Inn }G,\circ)\) and \((\operatorname{Aut} G,\circ)\) be the groups of all inner and all automorphisms of \((G,\cdot)\), and let \((\text{End }G,\circ)\) be the semigroup of all endomorphisms of \((G,\cdot)\). These three subsets of \(M(G)\) generate three subnear-rings \(I(G)\subseteq A(G)\subseteq E(G)\) of \((M(G),+,\circ)\). They are called the endomorphism near-rings.
In this article, we consider endomorphism near-rings of the holomorph of a cyclic \(p\)-group for odd prime number \(p\). By definition, the holomorph of a group \(A\) is the natural semidirect product of \(A\) and \(\operatorname{Aut} A\). It is easy to see that the holomorph \(H\) of a cyclic group of order \(2^n\) does not satisfy \(I(H)=E(H)\) for \(n\geq 2\).
We prove the following Theorem: Let \(p\) be an odd prime and \(G\) be the holomorph of a cyclic group of order \(p^n\). Then \(I(G)=A(G)=E(G)\) if and only if \(n\leq p\).