Group algebras with units satisfying an Engel identity (Q1851401)

The main result in this paper shows that, for a periodic group \(G\) and a field \(F\) of prime characteristic \(p>0\) the fact that the unit group \(U(FG)\) is \(n\)-Engel, for some \(n\), is equivalent to \(FG\) being \(n\)-Engel, for some \(n\), and that this happens if and only if \(G\) is a nilpotent group containing a normal subgroup \(A\) such that both \(G/A\) and \(A'\) are finite \(p\)-groups. The implication showing that \(FG\) being \(n\)-Engel gives that \(U(FG)\) \(n\)-Engel had already been proved by the present author and \textit{M. C. Wilson} in a more general context [in Proc. Am. Math. Soc. 127, No. 4, 973-976 (1999; Zbl 0915.16023)].