Picard groups and automorphism groups of integral group rings of metacyclic groups

Abstract

Let K be a Dedekind domain with quotient field K, and let A be an H order in a separable K-algebra A. A. Frohlich [4] introduced the locally free Picard group of A, which is denoted by LFP(,4). This is defined to be the group of all isomorphism types of invertible /i-/i-bimodules in A which are locally free as left n-modules. As usual, the locally free class group of ,4 is denoted by C(A). It is noted that if/i is commutative, then LFP(/I) z C(/i). Let Aut(/l) be the group of all automorphisms of n and let In(A) be the subgroup of Aut(/i) consisting of all inner automorphisms of/i. Let cent@) denote the center of /i. An automorphism f of /i is called a central automorphism iff(c) = c for any c C ccnt(/i). WC denote by Autccnt(/l) the subgroup of Aut(/l) consisting of all central automorphisms of A, and define Outcent = Autcent(/i)/In(/l). Fundamental properties of the groups LFP(A) and Outcent were systematically studied by Friihlich, Reiner and Ullom (4, 5 1. Let G be a finite group and let LG be its integral group ring. We denote by C, the cyclic group of order n, by D, the dihedral group of order 2n and by H, the quaternion group of order 4n. Frohlich, Reiner and Ullom computed LFP(ZG) and Outccnt(LG) when G = D,, II, or D,, p an odd prime. 286 0021.8693/82/08028~25%02.00/0