Class Groups of Integral Group Rings(x)
نویسنده
چکیده
Let A be an Ä-order in a semisimple finite dimensional /(-algebra, where K is an algebraic number field, and R is the ring of algebraic integers of K. Denote by C(A) the reduced class group of the category of locally free left A-lattices. Choose A= ZC, the integral group ring of a finite group G, and let A be a maximal Z-order in QG containing A. There is an epimorphism C(A)-C(A'), given by JIÍ —A' ®A M, for M a locally free A-lattice. Let D(A) be the kernel of this epimorphism; the groups D(A), C(A) and C(A ) are all finite. Our main theorem is that D(ZG) is a p-group whenever G is a p-group. This generalizes Frohlich's result for the case where G is an abelian p-group. Our proof uses some facts about the center F of QG, as well as information about reduced norms. We also calculate D(ZG) explicitly for G cyclic of order 2p, dihedral of order 2p, or the quaternion group. In these cases, the ring ZG can be conveniently described by a pullback diagram.
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