Detection theorems for K-theory and L-theory

Suppose G is a p-hyperelementary group and R is a commutative ring such that the order of G is a unit in R. Suppose J is either one of Quillen's K-theory functors or one of Wall's oriented L-theory functors. We show that J(RG) can be detected by applying J(R?) to the subquotients of G such that all normal abelian subgroups are cyclic. In 3.A.6 we show that such subquotients have a quite simple structure.We also show how to detect more general L-theory functors, in particular unoriented ones and those that arise in the study of codimension one submanifolds.