A transfer result for powerful Sylow subgroups (Q1904082)

By \(P^n\) we denote the subgroup generated by all \(n\)-th powers of elements of \(P\). A \(p\)-group \(P\) is called powerful if either \(p\) is odd, and \(P^p\geq P'\), or \(p=2\), and \(P^4\geq P'\). A \(p\)-group \(P\) is called regular if for every \(x,y\in P\) we have \((xy)^p\equiv x^py^p\bmod (H')^p\), where \(H=\langle x,y\rangle\). Let \(G\) be a finite group, let \(P\) be a Sylow \(p\)-subgroup, let \(N=N_G(P)\) be the normalizer of \(P\). A theorem of Wielandt [\textit{B. Huppert}, ``Endliche Gruppen, I'' (1979; Zbl 0412.20002), p. 447-452] asserts, that if \(P\) is a regular \(p\)-group then the maximal \(p\)-factor groups of \(G\) and \(N\) are isomorphic. The goal of the present paper is to prove the same conclusion under the assumption that \(P\) is a powerful group. The proof is based on the above-mentioned theorem of Wielandt.