Group algebras whose units satisfy a group identity. II

Let $K[G]$ be the group algebra of a torsion group $G$ over an infinite field $K$, and let $U=U(G)$ denote its group of units. A recent paper of A. Giambruno, S. K. Sehgal, and A. Valenti proved that if $U$ satisfies a group identity, then $K[G]$ satisfies a polynomial identity, thereby confirming a conjecture of Brian Hartley. Here we add a footnote to their result by showing that the commutator subgroup $G'$ of $G$ must have bounded period. Indeed, this additional fact enables us to obtain necessary and sufficient conditions for $U(G)$ to satisfy an identity.