Modular group algebras with centrally metabelian unit groups (Q6169179)

Let \(\mathrm{Syl}_p(G)\) be the Sylow \(p\)-subgroup of a group \(G\), i.e., the maximal \(p\)-subgroup of \(G\) and let \(G'\) be the first commutator subgroup of \(G\). The group \(G\) is called central metabelian if the second commutator subgroup \(G''\) of \(G\) is central in \(G\). Let \(FG\) be a group algebra of the group \(G\) over a field \(F\) of prime characteristic \(p\) and let \(U(FG)\) be the unit group of \(FG\). The group algebra \(FG\) is called modular if \(G\) has an element of order \(p\). The authors prove the following result. Theorem 1. Let \(F\) be a field of characteristic \(p>2\) and \(G\) a non abelian group such that \(FG\) is modular. Then \(U(FG)\) is central metabelian if and only if \begin{itemize} \item[1)] \(p=3\) and \(G'\) has order \(p\), \item[2)] \(p=3\) and \(G'=\mathrm{Syl}_p(G)\) is central, elementary of order \(p^3\), or \item[3)] \(p=5\) and \(G'=\mathrm{Syl}_p(G)\) is central of order \(p\). \end{itemize}