On iterated group actions and direct products (Q1969115)

Given a group \(G\), let it act on itself via conjugation. Consider the corresponding semidirect product \(G\propto G\). Then it is known that \(G\propto G\cong G\times G\) (this result has appeared as Exercise 7.10 in \textit{I. M. Isaacs'} ``Algebra; a graduate course'' (1993; Zbl 0805.00001)). In this paper the author expands this idea. Let \(G\) be a group and \(t\geq 2\) an integer. Consider the set \(G^t\), consisting of all \(t\)-tuples \((a,b,\ldots)\), where \(a,b,\ldots\in G\). We can equip \(G^t\) with an operation \(*\) by means of \[ (a,b,c,d,\ldots)*(\overline a,\overline b,\overline c,\overline d,\ldots)=(({^1 a})\overline a,({^{\overline a^{-1}} b})\overline b,({^{(\overline a\overline b)^{-1}} c})\overline c,({^{(\overline a\overline b\overline c)^{-1}} d})\overline d,\ldots), \] where \({^yx}=yxy^{-1}\). \(G^t\) has a group structure under the operation \(*\). Some properties of the group \(G^t\) are studied. These considerations give rise to the following theorem: Suppose that \(M\) and \(N\) are normal subgroups of \(G\). Let \(H=\{b\in G\mid\) there exists \(a\in M\) such that \(ab\in N\}\). Then \(H=MN\).