A note on factorized (finite) \(p\)-groups (Q1378836)

A well-known theorem of Kegel and Wielandt says that every finite product of two nilpotent subgroups is soluble. The author is interested in bounding the derived length of groups \(G=AB\) which are the product of two \(p\)-groups \(A\) and \(B\). The main result asserts that, if \(G\) is finite, and if the derived subgroups of \(A\) and \(B\) have orders \(p^m\geq p^n\) (resp.), then the derived length of \(G\) is at most \(m+2n+2\). This improves an earlier bound of Kazarin.