ISRN Algebra

Volume 2013 (2013), Article ID 638623, 4 pages

http://dx.doi.org/10.1155/2013/638623

Research Article

On Finite Nilpotent Matrix Groups over Integral Domains

Dmitry Malinin

Department of Mathematics, UWI, Mona Campus, Kingston 7, Jamaica

Received 31 October 2013; Accepted 19 November 2013

Academic Editors: I. Cangul and N. Jing

Copyright © 2013 Dmitry Malinin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 implies that for , , and for , . Therefore, is decomposable. This completes the proof of Proposition 2.

Proof of Theorem 1. Let us denote by the subgroup of all scalar matrices in , and let be the centre of .

If , we use Proposition 2 and induction on .

Let , and let the exponent of the group be equal to . Let be any element not contained in the centre of such that the image of in the factor group is contained in the centre of . Then we consider the homomorphism

given by . The kernel of is the set of all scalar matrices contained in . This kernel is not trivial since is nilpotent. The image is isomorphic to the factor group . Let . Then is a nonscalar diagonal matrix in the centre of , is also not a scalar matrix, and for any we have for some root of . If for the matrix the elements and are not zero, we obtain and , which implies immediately that if (as in case (i) of Theorem 1). In case (ii) of Theorem 1, if , we obtain and , but is impossible in the virtue of the condition that is not contained in . Hence we have , and is contained in , the centre of . This contradiction completes the proof of Theorem 1.

Proposition 3. Let be an ideal of a Dedekind ring of characteristic , let , and let be a matrix of finite order congruent to .

(i)If , then   for some integer . If , then contains a prime number and for some integer . In particular, a finite group of matrices congruent to is a -group.(ii)Let , and let be a prime ideal having ramification index with respect to , let , and let

Then ; in particular, any finite group of matrices congruent to is trivial if .

See [17, Lemma 1] for the proof of Proposition 3.

The following corollary can be immediately obtained from Proposition 3.

Corollary 4. Let be a number field of degree with the maximal order . The kernel of reduction of modulo an ideal of , containing a prime number , has no torsion if the norm .

Remark 5. Earlier Bürgisser obtained a similar result for ; see [18, Lemma 3.1].

Propositions 3 and 6 below can be used for estimating the orders of finite subgroups of using the reduction modulo some prime ideal . It is also possible to determine the structure of a -subgroup of having the maximal possible order with some modifications in the case . The theorems describing the maximal -subgroups of over fields can be found in [19]; in particular, it is proven that there is only one conjugacy class of maximal -subgroups of for ; see also [9, 20]. However, the equivalence of subgroups in over is a more subtle question. See [21, chapter 3], [22, 23] for the structure of finite linear groups (including the groups of small orders). See [15] for more details, proofs, and applications to determining torsion elements in the reduced projective class group.

As a corollary of Theorem 1 we can obtain the following proposition.

Proposition 6. Let be a finite extension, and let . Let , and let . Let be a finite subgroup of and for all . Then is conjugate in to an abelian group of diagonal matrices of exponent .

Proof of Proposition 6. Let us prove that is abelian of exponent . Let be a prime element of . Let , for some . Then , , and . It follows from Proposition 2 that and the same proposition shows that for any . First of all, is conjugate over to a group of triangular matrices, since is abelian and is a local ring; see [6, Theorem 73.9] and the remarks in [6, on page 493]. Following Theorem 1, let us prove that every is diagonalizable. We can describe explicitly the matrix such that

is a diagonal matrix for a triangular matrix of order which is congruent to . Indeed, let and

and let

for and , are appropriate -roots of 1. We consider

and we find the system of conditions for providing , the zero matrix. We have the following system of conditions:

The condition implies that , and we can find , sequentially using the results of the previous steps:

and so on. Now, using the induction on the degree we can find a matrix that transforms to a diagonal form as required.

The condition of Proposition 6 implies that the condition (i) of Theorem 1 holds true. Since is an abelian group of exponent this allows us to prove our claim over the ring .

Remark 7. Using the same argument for an algebraic number fields we can prove a similar result. Let be a Dedekind ring in , and let . Let , . Let be a finite subgroup of and for all . Then is conjugate in to an abelian group of diagonal matrices of exponent .

In this situation we can use the statement (81.20) in [CR] for proving the above result for the given Dedekind ring (compare also the proof of (81.20) and (75.27) in [6]).

However, in Proposition 6 the ramification index . Below there are two examples giving constructions of local field extensions and finite subgroups which are contained in the kernel of reduction of modulo ideals having ramification indices .

Example 8. For the following finite extension of local fields obtained via adjoining torsion points of elliptic curves, let be the ring of integers of with the maximal ideal . Consider an elliptic curve over with supersingular good reduction (see [24, Section 1.11]). Let be the field extension obtained by adjoining -torsion points of ; then the formal group associated with has a height of 2; its Hopf algebra is a free module of rank over and for the kernel of multiplication by , (see [25, 1.3 and Section 2]). Note that for some the ramification index ([24, page 275, Proposition 12]).

We can consider the group of -torsion points as -algebra homomorphisms from the Hopf algebra to the -algebra ; then , and the algebra is isomorphic to ; see [25, Section 2] and [26]. So there is a representation , and since is supersingular, the image of is contained in the kernel of reduction modulo .

The following example shows that the kernel of reduction of modulo a prime divisor of may contain -groups of any prescribed nilpotency class for extensions with large ramification; these groups are not abelian, and they are not diagonalizable in .

Example 9. Let us consider the following -group of nilpotency class , determined by generators and relations , , ; , , ; , where and is a suitable positive integer. Let be the abelian subgroup of generated by , and let denotes the character of given on the generators as follows: primitive -root of 1, , . The character together with the decomposition of into cosets with respect to : gives rise to an induced representation of . For the matrices having precisely one nonzero entry in the position equal to 1 we can define a matrix using the binomial coefficients :

Theorem 10. Let denote the extension of obtained by adjoining all roots , of -primary orders of 1, let be the uniformizing element of a finite extension such that , and let . Then for the representation of is a faithful, absolute, irreducible representation in by matrices congruent to . Moreover, such representations are pairwise nonequivalent over , and for the lower central series of all elements of are congruent to if the elements of are congruent to .

For the proof of Theorem 10 (which is constructive) see [17, 27]. Remark that the construction of Theorem 10 can be realized also over the integers of cyclotomic subextensions of and other global fields.

Acknowledgment

The author is grateful to the referees for useful suggestions.

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