We consider finite nilpotent groups of matrices over commutative rings. A general result concerning the diagonalization of matrix groups in the terms of simple conditions for matrix entries is proven. We also give some arithmetic applications for representations over Dedekind rings. 1. Introduction In this paper we consider representations of finite nilpotent groups over certain commutative rings. There are some classical and new methods for diagonalizing matrices with entries in commutative rings (see [1, 2]) and the classical theorems on diagonalization over the ring of rational integers originate from the papers by Minkowski; see [3–5]. We refer to [6–8] for the background and basic definitions. First we prove a general result concerning the diagonalization of matrix groups. This result gives a new approach to using congruence conditions for representations over Dedekind rings. The applications have some arithmetic motivation coming back to Feit [9] and involving various arithmetic aspects, for instance, the results by Bartels on Galois cohomologies [10] (see also [11–14] for some related topics) and Bürgisser [15] on determining torsion elements in the reduced projective class group or the results by Roquette [16]. Throughout the paper we will use the following notations. , , , , , , and denote the fields of complex and real numbers, rationals and -adic rationals, the ring of rational and -adic rational integers, and the ring of integers of a local or global field , respectively. denotes the general linear group over . denotes the degree of the field extension . denotes the unit matrix. is a diagonal matrix having diagonal components . denotes the order of a finite group . Theorem 1. Let be a commutative ring, which is an integral domain, and let be a finite nilpotent group indecomposable in . Let one suppose that every matrix is conjugate in to a diagonal matrix. Then any of the following conditions implies that is conjugate in to a group of diagonal matrices:(i)every matrix in has at least one diagonal element ,(ii) is not contained in , where is the identity matrix, and for any matrix in , there are 2 indices such that and . For the proof of Theorem 1 we need the following. Proposition 2. If the centre of a finite subgroup for a commutative ring , which is an integral domain, contains a diagonal matrix , then is decomposable. Proof. After a conjugation by a permutation matrix we can assume that where for and contains elements that equal , . For a matrix consider the system of linear equations determined by the conditions ; this immediately
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