Whitehead Groups of Finite Groups

In 1966, Milnor surveyed in this Bulletin [23] the concept of Whitehead torsion, focusing on the definition, topological significance and computation of Whitehead groups and their relationship to algebraic ^-theory and the congruence subgroup problem. As Milnor showed in that survey [23, Appendix 1], an affirmative solution to the congruence subgroup problem for algebraic number fields would imply that for any finite abelian group G, SKX(ZG) = 0; i.e. that the Whitehead group of a finite abelian group G is torsion-free. At that time the status of the congruence subgroup problem was uncertain [23, pp. 360, 416]; it was subsequently shown to have a negative solution by Bass, Milnor and Serre [7]. Nevertheless, until 1972 all finite abelian groups for which computations could be made had trivial SKl (cf. [5, p. 624]) and the question of whether these groups could be nontrivial remained open [6]. An intensive study of Milnor's K2"l\mcior o n discrete valuation rings [10] and the application of Mayer-Vietoris sequences in algebraic ^-theory led to the first examples of finite abelian groups with nontrivial SKX and have provided an algorithm for the computation of such SKx's in general. In addition, the first steps towards the computation of SKX(ZG) for nonabelian finite groups have been taken by several authors. It is my purpose to survey these techniques and computations, beginning where Milnor left off in 1966.1 will rely heavily on his article for background material; all unexplained notations and terminology should be sought there. If G is a finite group, its order is denoted \G\ and its abelianization, G. A finite field with q elements is denoted F^. The units of a ring A are denoted A*orU(A). I would like to thank Bruce Magurn, Keith Dennis and Michael Keating for their helpful comments.