The combinatorial derivation. (Q2871996)

Denote for each group \(G\) by \(2^G\) the family of all subsets of \(G\) and define \(\Delta(A)=\{g\in G:|gA\cap A|\geq\aleph_0\}\) for each subset \(A\) of \(G\). The mapping \(\Delta\colon 2^G\to 2^G\), \(A\mapsto\Delta(A)\) is called the combinatorial derivation. The \(\Delta\)-trajectory of a subset \(A\) of \(G\) is the sequence \(A,\Delta(A),\Delta^2(A),\ldots\).NEWLINENEWLINE The author shows that for any infinite group \(G\) and for each symmetric subset \(A\) containing the identity \(e_G\) of \(G\) there exists a subset \(X\) with \(\Delta(X)=A\). It is proved also that if \(G=A_1\cup\cdots\cup A_n\) is a partition of a group \(G\), then for some \(i\in\{1,\ldots,n\}\) there exists a finite subset \(F\) of \(G\) such that \(G=F\Delta(A_i)\). Let \(G\) be a countable group with the property that for each \(g\neq e_G\) the set \(\{x\in G:x^2=g\}\) is finite. Under these conditions \(\Delta\)-trajectories with special properties are constructed. It is proved, for instance, that for every natural number \(n\) there exists a periodic \(\Delta\)-trajectory of length \(n\).