The construction of the solutions of the generalized translation equation (Q1849463)

The well-known result of \textit{M. Hosszú} [Publ. Math. 10, 88-92 (1964; Zbl 0118.26402)], that if \(\langle G,\cdot\rangle\) is a binary group, \(\mu\) is an automorphism of \(\langle G,\cdot\rangle\) and \(a\in G\), \(\mu(a)=a\) and \(\mu^{n-1}(x)=axa^{-1}\) then the operation defined by \[ [x_1x_2\dots x_n]:=x_1\cdot\mu(x_2)\cdot\mu^2(x_3)\cdot{\dots}\cdot\mu^{n-1}(x_n)\cdot a \] is the \(n\)-group operation in \(G\). Hence the equation \[ \varphi(\varphi(\varphi(\dots(\varphi(\alpha,x_1),x_2),\dots),x_{n-1}),x_n)= \varphi(\alpha, x_1\cdot\mu(x_2)\cdot\mu^2(x_3)\cdot{\dots}\cdot\mu^{n-1}(x_n)\cdot a) \] with \(\varphi:\Gamma \times G\to G\), where \(\Gamma\) is an arbitrary set, represents the translation equation on an arbitrary \(n\)-group. In this paper the general construction of solutions of the equation \[ \varphi(\varphi(\varphi(\dots(\varphi(\alpha,x_1),x_2),\dots),x_{n-1}),x_n)= \varphi(\alpha, x_1\cdot\mu(x_2)\cdot\mu^2(x_3)\cdot{\dots}\cdot\mu^{n-2}(x_{n-1})\cdot x_n) \] is given, this is a translation equation on special \(n\)-groups which are obtained for \(a=e\), \(e\) is the unit element of \(G\). The equation (1) is a generalization of the classical translation equation \[ F:\Gamma \times G\rightarrow \Gamma, F(F(\alpha,x),y)=F(\alpha,xy). \]