Nilpotency class of symmetric units of group algebras

Let F be a field of odd prime characteristic p, G a group, U the group of units in the group algebra FG, and U⁺ the subgroup of U generated by the elements of U fixed by the anti-automorphism of FG which inverts all elements of G. It is known that U is nilpotent if G is nilpotent and the commutator subgroup G′ has p-power order, and then the nilpotency class of U is at most the order of G′; this bound is attained if and only if G′ is cyclic and not a Sylow subgroup of G. Adalbert Bovdi and János Kurdics proved the 'if' part of this last statement by exhibiting a nontrivial commutator of the relevant weight. For the special case when G is a nonabelian torsion group (so G′ cannot possibly be a Sylow subgroup), the present paper identifies such a commutator in U⁺, showing (Theorem 1) that the same bound is attained even by the nilpotency class of this subgroup. We do not know what happens when G′ is not a Sylow subgroup but G is not torsion. It can happen that U⁺ is nilpotent even though U is not. The torsion groups G which allow this are known (from the work of Gregory T. Lee) to be precisely the direct products of a finite p-group P, a quaternion group Q of order 8, and an elementary abelian 2-group. Theorem 2: in this case, the nilpotency class of U⁺ is strictly smaller than the nilpotency index of the augmentation ideal of the group algebra FP, and this bound is attained whenever P is a powerful p-group. The nonabelian group P of order 27 and exponent 3 is not powerful, yet the G = P × Q formed with this P also leads to a U⁺ attaining the general bound, so here a necessary and sufficient condition remains elusive.