Let GG be a group, FF a field of characteristic p≥0p≥0, and 𝒰(FG) the unit group of the group algebra FG. In this paper, among other results, we show that if either (1) FG satisfies a non-matrix polynomial identity, or (2) G is locally finite, F is infinite and 𝒰(FG) is an Engel-by-finite group, then the p-elements of G form a (normal) subgroup P and G/P is abelian (here, of course, P=1 if p=0).