Reduction of [formula omitted] for finite groups with normal abelian Sylow 2-subgroup

Abstract

Let F be a finite group with a Sylow 2-subgroup S that is normal and abelian. Using hyperelementary induction and cartesian squares, we prove that Cappell’s unitary nilpotent groups UNil ∗ ( Z [ F ] ; Z [ F ] , Z [ F ] ) have an induced isomorphism to the quotient of UNil ∗ ( Z [ S ] ; Z [ S ] , Z [ S ] ) by the action of the group F / S . In particular, any finite group F of odd order has the same UNil -groups as the trivial group. The broader scope is the study of the L -theory of virtually cyclic groups, based on the Farrell–Jones isomorphism conjecture. We obtain partial information on these UNil when S is a finite abelian 2-group and when S is a special 2-group.