Meta-abelian unit groups of group algebras are usually abelian (Q1176697)

Let \(G\) be a finite group and let \(F\) be a field of characteristic \(p>0\). Let \(U(FG)\) be the unit group of the group algebra \(FG\). It is known that, for \(p\geq 5\) \(U(FG)\) is soluble if and only if \(G/O_ p(G)\) is abelian. It is here shown that, for \(p\geq 5\), \(U(FG)\) is metabelian if and only if \(G\) is abelian.