# On Finite Nilpotent Matrix Groups over Integral Domains

Authors: Dmitry Malinin
Journal: ISRN Algebra
 (2013)

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## 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.


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## Body

## 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.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.
LetA 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 in GLn(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:(i)
every matrixg=[gij]i,j in G has at least one diagonal element gii≠0,(ii)
-In is not contained in G, where In is the identity n×n matrix, and for any matrix g=[gij]i,j in G, there are 2 indices i,j such that gij≠0 and gji≠0.For the proof of Theorem1 we need the following.Proposition 2.
If the centre of a finite subgroupG⊂GLn(A) for a commutative ring A, which is an integral domain, contains a diagonal matrix d≠In, then G is decomposable.Proof.
After a conjugation by a permutation matrix we can assume that(1)d=diag(d1,…,d1,d2,…d2,…,di,…,di,…,dk,…,dk),
where di≠dj for i≠j and d contains ti elements that equal di, i=1,…,k. For a matrix g=[gij]i,j∈G consider the system of linear equations determined by the conditions gd=dg; this immediately implies that gij=0 for i≤t1, j>t1, and gkm=0 for m≤t1, k>t1. Therefore, G is decomposable. This completes the proof of Proposition 2.Proof of Theorem1.
Let us denote byD the subgroup of all scalar matrices in G, and let Z be the centre of G.
IfD≠Z, we use Proposition 2 and induction on n.
LetD=Z, and let the exponent of the group Z be equal to t. Let g0∈G be any element not contained in the centre Z of G such that the image of g0 in the factor group G/Z is contained in the centre of G/Z. Then we consider the homomorphism
(2)ψ:G′⟶GLnt(A),
given byψ(g)=g⊗t. The kernel of ψ is the set D of all scalar matrices contained in G. This kernel is not trivial since G is nilpotent. The image ψ(G) is isomorphic to the factor group G/D. Let g0=diag(d1,…,dn). Then ψ(g0) is a nonscalar diagonal matrix in the centre of ψ(G), g0 is also not a scalar matrix, and for any g∈G we have gg0=g0gζ for some root of  1ζ=ζ(g). If for the matrix g=[gnk]n,k the elements gij and gji are not zero, we obtain dj=diζ and di=djζ, which implies immediately that ζ=1 if i=j (as in case (i) of Theorem 1). In case (ii) of Theorem 1, if i≠j, we obtain ζ2=1 and ζ=±1, but ζ=-1 is impossible in the virtue of the condition that -In is not contained in G. Hence we have ζ=1, and g0 is contained in Z, the centre of G. This contradiction completes the proof of Theorem 1.Proposition 3.
Letℐ be an ideal of a Dedekind ring S of characteristic χ, let {0}≠ℐ≠S, and let g be a n×n matrix of finite order congruent to In(modℐ).(i)
Ifχ=p>0, thengpj=Infor some integerj. Ifχ=0, thenIcontains a prime numberpandgpi=Infor some integeri. In particular, a finite group of matrices congruent toIn(modℐ)is ap-group.(ii)
Letχ=0, and letℐ=𝔭be a prime ideal having ramification indexewith respect top, letg≡In(mod𝔭r), and let(3)λpi-1(p-1)⩽er<pi(p-1),i⩾0,λ=min{1,i}.
Thengpi=In; in particular, any finite group of matrices congruent to In(mod𝔭t) is trivial if e<r(p-1).See [17, Lemma 1] for the proof of Proposition 3.The following corollary can be immediately obtained from Proposition3.Corollary 4.
LetK be a number field of degree d=[K:ℚ] with the maximal order OK. The kernel of reduction of GLn(OK) modulo an ideal ℐ of OK, containing a prime number p, has no torsion if the norm NK/ℚ(ℐ)>pd/(p-1).Remark 5.
Earlier Bürgisser obtained a similar result forNK/ℚ(ℐ)>2d; see [18, Lemma 3.1].Propositions3 and 6 below can be used for estimating the orders of finite subgroups of GLn(𝕆K) using the reduction modulo some prime ideal 𝔭⊂OK. It is also possible to determine the structure of a p-subgroup of GLn(OK) having the maximal possible order with some modifications in the case p=2. The theorems describing the maximal p-subgroups of GLn(K) over fields can be found in [19]; in particular, it is proven that there is only one conjugacy class of maximal p-subgroups of GLn(K) for p>2; see also [9, 20]. However, the equivalence of subgroups in GLn(OK) over OK 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 Theorem1 we can obtain the following proposition.Proposition 6.
LetK/ℚp be a finite extension, and let ζp∈OK. Let p=𝔭e, and let e=p-1. Let G be a finite subgroup of GLn(OK) and g≡In(mod𝔭) for all g∈G. Then G is conjugate in GLn(OK) to an abelian group of diagonal matrices of exponent p.Proof of Proposition6.
Let us prove thatG is abelian of exponent p. Let π be a prime element of OK. Let g1=In+πB1, g2=In+πB2 for some g1,g2∈G. Then gi-1≡In-πBi(modπ2),   i=1,2, and h=g1g2g1-1g2-1≡In(modπ2). It follows from Proposition 2 that h=In and the same proposition shows that gp=In for any g∈G. First of all, G is conjugate over OK to a group of triangular matrices, since G is abelian and OK 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 g∈G is diagonalizable. We can describe explicitly the matrix M such that
(4)M-1gM=diag(λ1,λ2,…,λn)
is a diagonal matrix for a triangular matrix g of order p which is congruent to In(mod𝔭). Indeed, let g∈G and
(5)g=|ζ(1)It1P21…Pk10ζ(2)It2…Pk2⋮⋱⋮0⋯ζ(k)Itk|,
and let
(6)S=|It10…A10It2…A2⋮⋱⋮0⋯Itk|,
for t1+t2+⋯+tk=n and t1≤t2≤⋯≤tk,ζ(i), i=1,2,…,k are appropriate p-roots of 1. We consider
(7)S-1gS=|ζ(1)It1*…Mk10ζ(2)It2…Mk2⋮⋱⋮0⋯ζ(k)Itk|,
and we find the system of conditions for providing Mki=0ti,tk, the zero ti×tk matrix. We have the following system of conditions:
(8)ζ(1)(1-ζ(k)ζ(1)-1)A1+P21A2+⋯+Pk-11Ak-1+Pk1=0t1,tk,⋯ζ(k-2)Ak-2(1-ζ(k)ζ(k-2)-1)+Pk-1k-2Ak-1+Pkk-2=0tk-2,tk,ζ(k-1)Ak-1(1-ζ(k)ζ(k-1)-1)+Pkk-1=0tk-1,tk.
The conditiong≡In(mod𝔭) implies that Pij≡0tjti(mod𝔭), and we can find Ai, 1≤i≤k-1 sequentially using the results of the previous steps:
(9)Ak-1=-Pkk-1ζ(k-1)(1-ζ(k)ζ(k-1)-1),Ak-2=-(Pkk-2+Pk-1k-2Ak-1)ζ(k-2)(1-ζ(k)ζ(k-2)-1),Ak-3=-(Pkk-3+Pk-1k-3Ak-1+Pk-2k-3Ak-2)ζ(k-3)(1-ζ(k)ζ(k-3)-1),
and so on. Now, using the induction on the degree n we can find a matrix M that transforms g to a diagonal form as required.
The conditiong≡In(mod𝔭) of Proposition 6 implies that the condition (i) of Theorem 1 holds true. Since G is an abelian group of exponent p this allows us to prove our claim over the ring OK.Remark 7.
Using the same argument for an algebraic number fieldsK/ℚ we can prove a similar result. Let O be a Dedekind ring in K, and let ζp∈O. Let p=𝔭e, e=p-1. Let G be a finite subgroup of GLn(O) and g≡In(mod𝔭) for all g∈G. Then G is conjugate in GLn(O) to an abelian group of diagonal matrices of exponent p.
In this situation we can use the statement (81.20) in [CR] for proving the above result for the given Dedekind ringO (compare also the proof of (81.20) and (75.27) in [6]).
However, in Proposition6 the ramification index e=p-1. Below there are two examples giving constructions of local field extensions K/ℚp and finite subgroups G⊂GLn(OK) which are contained in the kernel of reduction of GLn(OK) modulo ideals having ramification indices e>p-1.Example 8.
For the following finite extensionK/ℚp of local fields obtained via adjoining torsion points of elliptic curves, let OK be the ring of integers of K with the maximal ideal 𝔭. Consider an elliptic curve E over ℤp with supersingular good reduction (see [24, Section 1.11]). Let K/ℚp be the field extension obtained by adjoining p-torsion points of E; then the formal group associated with E has a height of 2; its Hopf algebra OA is a free module of rank p2 over ℤp and for the kernel Ep of multiplication by p, |Ep|=p2 (see [25, 1.3 and Section 2]). Note that for some E the ramification index e=e(K/ℚp)=p2-1 ([24, page 275, Proposition 12]).
We can consider the groupG of p-torsion points as ℤp-algebra homomorphisms from the Hopf algebra OA to the ℤp-algebra OK; then G=Homℤp(OA,OK), and the algebra OA is isomorphic to ℤp[X]/(c1X+c2X2+⋯+Xp2); see [25, Section 2] and [26]. So there is a representation v:G→GLp2(OK), and since E is supersingular, the image of v is contained in the kernel of reduction modulo 𝔭.
The following example shows that the kernel of reduction ofGLn(OK) modulo a prime divisor of p may contain p-groups of any prescribed nilpotency class l>1 for extensions K/ℚp with large ramification; these groups are not abelian, and they are not diagonalizable in GLn(OK).Example 9.
Let us consider the followingp-group of nilpotency class l, determined by generators a,b1,…,bl and relations bip=1, bibj=bjbi, i=1,2,…,l; ab1=b1a, bi-1=biabi-1a-1, i=1,2,…,l; an=1, where n=pt≥l>pt-1 and t is a suitable positive integer. Let H be the abelian subgroup of G generated by b1,…,bl, and let χ denotes the character of H given on the generators as follows: χ(b1)=ζp-a primitive p-root of 1, χ(bi)=1, i=2,…,l. The character χ together with the decomposition of G into cosets with respect to H: G=1·H+a·H+⋯+an-1·H gives rise to an induced representation R=IndχHG of G. For the n×n matrices eij having precisely one nonzero entry in the position (i,j) equal to 1 we can define a n×n matrix using the binomial coefficients (n-ji-j):
(10)C=Σn≥i≥j≥1(-1)i-j(n-ji-j)eij.Theorem 10.
Letℚp(ζp∞) denote the extension of ℚp obtained by adjoining all roots ζpi, i=1,2,3,… of p-primary orders of 1, let π be the uniformizing element of a finite extension K/ℚp such that K⊂ℚp(ζp∞), and let D=diag(1,π,π2,…,πn-1). Then for g∈G the representation Rπ(g)=D-1C-1R(g)CD of G is a faithful, absolute, irreducible representation in GLn(OK) by matrices congruent to In(modπ). Moreover, such representations are pairwise nonequivalent over Oℚp(ζp∞), and for the lower central series G=Gl⊃Gl-1⊃⋯⊃G0={In} of G all elements of Rπ(Gl-i+1) are congruent to In(modπiw) if the elements of Rπ(G) are congruent to In(modπw).For the proof of Theorem10 (which is constructive) see [17, 27]. Remark that the construction of Theorem 10 can be realized also over the integers of cyclotomic subextensions K⊂ℚ(ζp∞)=⋃i=1∞‍ℚ(ζpi) of ℚ and other global fields.



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