Variations on the Baer–Suzuki theorem

Abstract

The Baer–Suzuki theorem says that if \(p\) is a prime, \(x\) is a \(p\)-element in a finite group \(G\) and \(\langle x, x^g \rangle \) is a \(p\)-group for all \(g \in G\), then the normal closure of \(x\) in \(G\) is a \(p\)-group. We consider the case where \(x^g\) is replaced by \(y^g\) for some other \(p\)-element \(y\). While the analog of Baer–Suzuki is not true, we show that some variation is. We also answer a closely related question of Pavel Shumyatsky on commutators of conjugacy classes of \(p\)-elements.