Fine structure of class groups of cyclic p-groups

Abstract

In this paper we study the module structure of the locally free class group CZ(zCP,+l) of the cyclic p-group CDncl of order pn+l, p prime (compare earlier results [2, 3, 5, 6, 131). The standard involution c of CD,+, extends to ZC,,+, and acts on the class group and the kernel group D(ZC,,+1); it is useful to look at the symmetric and skew symmetric elements with respect to c. As modules, if (P) is the class in CZ(ZC,,+l) of a locally free module P, then (P)” is the inverse of the class of the contragredient module of P (see [4]). More generally, the whole automorphism group of CD,+, acts on the class group. For p odd, one then considers the eigenspaces %‘I (0 < i < p 2) of thep-Sylow subgroup of the class group with respect to the action of the subgroup A of Aut C,,+, of order p 1; these eigenspaces afford the A-decomposition. Further one defines a A= Z,[[T]]module structure on each eigenspace. In particular the iD(ZCP,+,) contain a large cyclic A-submodule iv, . A generator of iTfn is given explicitly and the A-annihilator of iV, is determined for odd i. Similarly V, is completely determined if p = 2. In Sections 2,3 results from the theory of cyclotomic fields are applied to prove firstly that the restriction map CZ(ZC,,+l) + CZ(ZC,,) is surjective and the induction map CZ(ZC,,)+ CZ(ZC,,+l)is injective. Secondly we prove the map CZ(ZC,n+1) -+ CZ (maximal order of QC Pn+~ is not split for p a properly irregular ) prime, n 3 1. Finally a counterexample to an assertion of [5] is given, and as an example the p-Sylow subgroup of CZ(ZC, ) 2 is computed exactly for p < 30,000.