87_14 - Invent.math. 87, 253-302 (1987) /n/2e/o~/8s...

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Invent. math. 87, 253-302 (1987) /n/2e////o~/8s mathematicae Springer-Verlag 1987 SK I of finite abelian groups, II Roger C. Alperin 1., R. Keith Dennis 2., R. Oliver 3, and Michael R. Stein 4. t Department of Mathematics, The University of Oklahoma, Norman, Oklahoma 73019, USA 2 Department of Mathematics, Cornell University, Ithaca, New York 14853, USA 3 Mathematisk Institut, Aarhus Universitet, Ny Munkegade, DK-8000 Aarhus C, Denmark 4 Department of Mathematics, Northwestern University, Evanston, Illinois 60201, USA Introduction This paper is a continuation of [A-D-S] in which we laid the theoretical foundation for computing the torsion, SKI(ZIG]), in the Whitehead group of a finite abelian group G. In that paper we showed how SKI(Z[G]) can be computed from a knowledge of SKI(Z[Gv]) for the Sylow p-subgroups of G, with some complications for p=2. In this paper we eliminate those com- plications, showing that when H is an abelian 2-group and (9 is a totally imaginary ring of integers in which 2 is unramified, then SK, ((9 [HI) ~ SK, (Z [H] ~ SK, ((9 [H/H2]), and we calculate the second summand. A second foundational result in [A-D-S] described SKI(TI[G]) as the cokernel of a map between the Kz's of certain finite rings. More specifically, for a finite abelian p-group G, SKI(Z[G]) is the cokernel of the homomor- phism q~s: K z (Z [G]/(pk)) ~ n K 2 (Z Ix]/(pk)) where the product is taken over certain equivalence classes of rational charac- ters Z of G, Z[X] is the ring of cyclotomic integers generated by Im(x), and k is large enough (e.g., pk> IG[). (This description requires some minor modification when p=2.) Now it follows from [D-S] that K2(Z[z]/(pk)) is isomorphic to Im(x ) under a homomorphism induced by the norm residue symbol, and this isomorphism is given explicitly by a formula of Artin and Hasse [A-H] involving the p-adic logarithm. The key idea of this paper, which is developed in w 1, is to use the integral logarithm of Taylor and Oliver to give a simple combinatorial description of Im(~os) as a subgroup of//Im(x) (Propositions 1.5 and 1.7). * Research supported by the National Science Foundation
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254 R.C. Alperin et al. In w we use this new description to compute SKI(ZIG]) for elementary abelian p-groups (the rank 3 case for p> 3 gave the first examples of non-trivial SKi's [A-D-So] ). We also compute SK1((9 [G]) for elementary abelian 2-groups G and rings of algebraic integers (9 in which 2 is unramified. We had previously shown [A-D-S, Theorem 3.4] that SK~((9[G]) vanishes if G is cyclic and the primes dividing its order do not ramify in (9. In w we generalize this result by removing the restriction on ramification. It then follows from induction theorems of Dress [Dr] and Oliver [OIII] that SK~((9[G])= 1 when G is metacyclic and (9 is any ring of algebraic integers; we prove, conversely, that metacyclic groups are the only finite groups with this property. In w we compute explicitly the exponent of SKx(Z[G]) for a finite abelian group G (Theorem 4.8) and we determine that the only such groups with SK~(Z[G])=I are those known prior to 1971: elementary abelian 2-groups, and groups all of whose Sylow p-subgroups are either cyclic or of type (p,p"). Finally, in w 5 we present a selection of further computations. The aim here
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87_14 - Invent.math. 87, 253-302 (1987) /n/2e/o~/8s...

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