Bounding the exponent of a verbal subgroup.
Let \(w=w(x_1,x_2,\ldots,x_m)\) be a group word and \(G\) be a group; the verbal subgroup of \(G\) defined by \(w\) is \(w(G)=\langle w(g_1,g_2,\ldots,g_m)\mid g_1,g_2,\ldots,g_m\in G\rangle\). In the paper under review the authors deal with the following conjecture. If \(G\) is a finite group in which any nilpotent subgroup generated by \(w\)-values has exponent dividing \(e\), then the exponent of \(w(G)\) is bounded in terms of \(w\) and \(e\) only. In Theorem A the previous conjecture is proved to be true in the case where \(w\) is either the \(n\)-th Engel word or the word \([x^r,y_1,y_2,\ldots,y_n]\). In Theorem B the authors are able to prove that for any positive integer \(e\) there exists a number \(k=k(e)\) such that if \(w\) is a word and \(G\) is a finite group in which any nilpotent subgroup generated by products of \(k\) values of the word \(w\) has exponent dividing \(e\), then the exponent of \(w(G)\) is bounded in terms of \(w\) and \(e\) only.
scientific article; zbMATH DE number 6363178
scientific article