On Finite Nilpotent Matrix Groups over Integral Domains

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 wewill use the following notations. C, R, Q, Qp, Z, Zp, and OK denote the fields of complex and real numbers, rationals and p-adic rationals, the ring of rational and p-adic rational integers, and the ring of integers of a local or global field K, respectively. GLn(R) denotes the general linear group over R. [E : F] denotes the degree of the field extension E/F. Im denotes the unit m × m matrix. diag (d1, d2, . . . , dm) is a diagonal matrix having diagonal components d1, d2, . . . , dn. |G| denotes the order of a finite group G. Theorem 1. Let A be a commutative ring, which is an integral domain, and let G ⊂ GLn(A) be a finite nilpotent group indecomposable in GLn(A). Let one suppose that every matrix g ∈ G is conjugate inGLn(A) to a diagonal matrix.Then any of the following conditions implies that G is conjugate in GLn(A) to a group of diagonal matrices:


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][4][5].We refer to [6][7][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][12][13][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.C, R, Q, Q  , Z, Z  , 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.GL  () denotes the general linear group over .[ : ] denotes the degree of the field extension /.  denotes the unit  ×  matrix.diag ( where   ̸ =   for  ̸ =  and  contains   elements that equal   ,  = 1, . . ., .For a matrix  = [  ] , ∈  consider the system of linear equations determined by the conditions  = ; this immediately implies that   = 0 for  ≤  1 ,  >  1 , and   = 0 for  ≤  1 ,  >  1 .Therefore,  is decomposable.This completes the proof of Proposition 2.

ISRN Algebra
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  0 ∈  be any element not contained in the centre  of  such that the image of  0 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  0 = diag( 1 , . . .,   ).Then ( 0 ) is a nonscalar diagonal matrix in the centre of (),  0 is also not a scalar matrix, and for any  ∈  we have  0 =  0  for some root of 1 = ().If for the matrix  = [  ] , the elements   and   are not zero, we obtain   =    and   =   , which implies immediately that  = 1 if  =  (as in case (i) of Theorem 1).In case (ii) of Theorem 1, if  ̸ = , we obtain  2 = 1 and  = ±1, but  = −1 is impossible in the virtue of the condition that −  is not contained in .Hence we have  = 1, and  0 is contained in , the centre of .This contradiction completes the proof of Theorem 1.

Proposition 3. Let I be an ideal of a Dedekind ring 𝑆 of characteristic 𝜒, let {0} ̸
= I ̸ = , and let  be a  ×  matrix of finite order congruent to   (mod I).
then  contains a prime number  and    =   for some integer .In particular, a finite group of matrices congruent to   (mod I) is a -group.
(ii) Let  = 0, and let I = p be a prime ideal having ramification index  with respect to , let  ≡   (mod p  ), and let Then    =   ; in particular, any finite group of matrices congruent to   (mod p  ) is trivial if  < ( − 1).
See [17, Lemma 1] for the proof of Proposition 3. The following corollary can be immediately obtained from Proposition 3. Propositions 3 and 6 below can be used for estimating the orders of finite subgroups of GL  (O  ) using the reduction modulo some prime ideal p ⊂   .It is also possible to determine the structure of a -subgroup of GL  (  ) having the maximal possible order with some modifications in the case  = 2.The theorems describing the maximal subgroups of GL  () over fields can be found in [19]; in particular, it is proven that there is only one conjugacy class of maximal -subgroups of GL  () for  > 2; see also [9,20].However, the equivalence of subgroups in GL  (  ) 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 /Q  be a finite extension, and let   ∈   .Let  = p  , and let  =  − 1.Let  be a finite subgroup of   (  ) and  ≡   (mod p) 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 ).It follows from Proposition 2 that ℎ =   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 and we find the system of conditions for providing    = 0   ,  , the zero   ×   matrix.We have the following system of conditions: The condition  ≡   (mod p) implies that    ≡ 0     (mod p), and we can find   , 1 ≤  ≤  − 1 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  ≡   (mod p) 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 /Q we can prove a similar result.Let  be a Dedekind ring in , and let   ∈ .Let  = p  ,  =  − 1.Let  be a finite subgroup of GL  () and  ≡   (mod p) for all  ∈ .Then  is conjugate in GL  () 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  =  − 1. Below there are two examples giving constructions of local field extensions /Q  and finite subgroups  ⊂ GL  (  ) which are contained in the kernel of reduction of GL  (  ) modulo ideals having ramification indices  >  − 1.
Example 8.For the following finite extension /Q  of local fields obtained via adjoining torsion points of elliptic curves, let   be the ring of integers of  with the maximal ideal p.Consider an elliptic curve  over Z  with supersingular good reduction (see [24,Section 1.11]).Let /Q  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  2 over Z  and for the kernel   of multiplication by , |  | =  2 (see [25,1.3 and Section 2]).Note that for some  the ramification index  = (/Q  ) =  2 − 1 ([24, page 275, Proposition 12]).
We can consider the group  of -torsion points as Z algebra homomorphisms from the Hopf algebra   to the Z  -algebra   ; then  = Hom Z  (  ,   ), and the algebra ); see [25, Section 2] and [26].So there is a representation V :  → GL  2 (  ), and since  is supersingular, the image of V is contained in the kernel of reduction modulo p.
The following example shows that the kernel of reduction of GL  (  ) modulo a prime divisor of  may contain groups of any prescribed nilpotency class  > 1 for extensions /Q  with large ramification; these groups are not abelian, and they are not diagonalizable in GL  (  ).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  ⊂ Q(  ∞ ) = ⋃ ∞ =1 Q(   ) of Q and other global fields.

Corollary 4 .
Let  be a number field of degree  = [ : Q] with the maximal order   .The kernel of reduction of   (  ) modulo an ideal I of   , containing a prime number , has no torsion if the norm  /Q (I) >  /(−1) .Remark 5. Earlier Bürgisser obtained a similar result for  /Q (I) > 2  ; see[18, Lemma 3.1].