Factor morphisms of nilpotent groups (Q1177454)

Let \(G\) be a nilpotent group and \(\xi_ 0,\xi_ 1,\xi_ 2,\dots\) is the upper central series of \(G\). The mapping \(\phi: G\to G\) is called a factor-morphism of \(G\) if for all elements \(u,v\in G\), where \([u,v]\in \xi_ i\), the following properties are true: 1) \((uv)^{\phi}(u^ \phi v^ \phi)^{-1}\in \xi_ i\); 2) \((u^ v)^ \phi=(u^ \phi)^ v\). Let \(\Phi(G)\) denote a set of all factor-morphisms of \(G\). It is easy to see that a factor-morphism \(\alpha\) of abelian group \(G\) is an endomorphism of \(G\). It is well known that a set \(E(G)\) of all endomorphisms of an abelian group \(G\) is a commutative ring with identity element under the operations: (3) \(g^{\alpha+\beta}=g^ \alpha g^ \beta\), \(g^{\alpha\beta}=(g^ \alpha)^ \beta\), where \(g\in G\), \(\alpha,\beta\in E(G)\). It is shown that the set \(\Phi(G)\) under the operations (3) is an associative ring with identity element. Some natural properties of \(\Phi(G)\) are established. A nilpotent group \(G\) is called a flat group if the second group of symmetric cohomologies \(H^ 2_ S(G/Z,Z)\) is trivial, where \(Z\) is a centre of \(G\). It is shown that a nilpotent group \(G\) with divisible centre is a flat group. It is proved that the flat group \(G\) is a direct product of proper subgroups if and only if the ring \(\Phi(G)\) is a direct sum of proper right ideals.