Research Article On Finite Nilpotent Matrix Groups over ...ISRN Algebra References [] W. C. Brown,...

Hindawi Publishing CorporationISRN AlgebraVolume 2013 Article ID 638623 4 pageshttpdxdoiorg1011552013638623

Research ArticleOn Finite Nilpotent Matrix Groups over Integral Domains

Dmitry Malinin

Department of Mathematics UWI Mona Campus Kingston 7 Jamaica

Correspondence should be addressed to Dmitry Malinin dmaliningmailcom

Received 31 October 2013 Accepted 19 November 2013

Academic Editors I Cangul and N Jing

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

We consider finite nilpotent groups of matrices over commutative rings A general result concerning the diagonalization of matrixgroups in the terms of simple conditions for matrix entries is proven We also give some arithmetic applications for representationsover Dedekind rings

1 Introduction

In this paper we consider representations of finite nilpotentgroups over certain commutative ringsThere are some classi-cal and new methods for diagonalizing matrices with entriesin commutative rings (see [1 2]) and the classical theoremson diagonalization over the ring of rational integers originatefrom the papers by Minkowski see [3ndash5] We refer to [6ndash8] for the background and basic definitions First we provea general result concerning the diagonalization of matrixgroupsThis result gives a new approach to using congruenceconditions for representations over Dedekind rings Theapplications have some arithmetic motivation coming backto Feit [9] and involving various arithmetic aspects forinstance the results by Bartels on Galois cohomologies [10](see also [11ndash14] for some related topics) and Burgisser [15] ondetermining torsion elements in the reduced projective classgroup or the results by Roquette [16]

Throughout the paper wewill use the following notationsC R Q Q119901 Z Z119901 and 119874119870 denote the fields of complexand real numbers rationals and 119901-adic rationals the ring ofrational and 119901-adic rational integers and the ring of integersof a local or global field 119870 respectively GL119899(119877) denotes thegeneral linear group over 119877 [119864 119865] denotes the degree ofthe field extension 119864119865 119868119898 denotes the unit 119898 times 119898 matrixdiag (1198891 1198892 119889119898) is a diagonal matrix having diagonalcomponents 1198891 1198892 119889119899 |119866| denotes the order of a finitegroup 119866

Theorem 1 Let 119860 be a commutative ring which is an integraldomain and let 119866 sub 119866119871119899(119860) be a finite nilpotent groupindecomposable in 119866119871119899(119860) Let one suppose that every matrix119892 isin 119866 is conjugate in119866119871119899(119860) to a diagonal matrixThen any ofthe following conditions implies that 119866 is conjugate in 119866119871119899(119860)

to a group of diagonal matrices(i) every matrix 119892 = [119892119894119895]119894119895 in 119866 has at least one diagonal

element 119892119894119894 = 0(ii) minus119868119899 is not contained in 119866 where 119868119899 is the identity 119899 times 119899

matrix and for any matrix 119892 = [119892119894119895]119894119895 in 119866 there are2 indices 119894 119895 such that 119892119894119895 = 0 and 119892119895119894 = 0

For the proof of Theorem 1 we need the following

Proposition 2 If the centre of a finite subgroup 119866 sub 119866119871119899(119860)

for a commutative ring119860 which is an integral domain containsa diagonal matrix 119889 = 119868119899 then 119866 is decomposable

Proof After a conjugation by a permutation matrix we canassume that119889 = diag (1198891 1198891 1198892 1198892 119889119894 119889119894 119889119896 119889119896)

(1)where 119889119894 = 119889119895 for 119894 = 119895 and 119889 contains 119905119894 elements that equal 119889119894119894 = 1 119896 For a matrix 119892 = [119892119894119895]119894119895 isin 119866 consider the systemof linear equations determined by the conditions 119892119889 = 119889119892this immediately implies that 119892119894119895 = 0 for 119894 le 1199051 119895 gt 1199051 and119892119896119898 = 0 for 119898 le 1199051 119896 gt 1199051 Therefore 119866 is decomposableThis completes the proof of Proposition 2

2 ISRN Algebra

Proof of Theorem 1 Let us denote by 119863 the subgroup of allscalar matrices in 119866 and let 119885 be the centre of 119866

If119863 =119885 we use Proposition 2 and induction on 119899Let119863 = 119885 and let the exponent of the group119885 be equal to

119905 Let1198920 isin 119866 be any element not contained in the centre119885 of119866such that the image of 1198920 in the factor group119866119885 is containedin the centre of 119866119885 Then we consider the homomorphism

120595 1198661015840997888rarr GL119899119905 (119860) (2)

given by 120595(119892) = 119892otimes119905

The kernel of 120595 is the set 119863 of all scalarmatrices contained in 119866 This kernel is not trivial since 119866 isnilpotent The image 120595(119866) is isomorphic to the factor group119866119863 Let 1198920 = diag(1198891 119889119899) Then 120595(1198920) is a nonscalardiagonal matrix in the centre of 120595(119866) 1198920 is also not a scalarmatrix and for any 119892 isin 119866 we have 1198921198920 = 1198920119892120577 for someroot of 1120577 = 120577(119892) If for the matrix 119892 = [119892119899119896]119899119896 the elements119892119894119895 and 119892119895119894 are not zero we obtain 119889119895 = 119889119894120577 and 119889119894 = 119889119895120577which implies immediately that 120577 = 1 if 119894 = 119895 (as in case(i) of Theorem 1) In case (ii) of Theorem 1 if 119894 = 119895 we obtain1205772= 1 and 120577 = plusmn1 but 120577 = minus1 is impossible in the virtue of the

condition thatminus119868119899 is not contained in119866 Hencewe have 120577 = 1and 1198920 is contained in 119885 the centre of 119866 This contradictioncompletes the proof of Theorem 1

Proposition 3 Let I be an ideal of a Dedekind ring 119878 ofcharacteristic 120594 let 0 =I = 119878 and let 119892 be a 119899 times 119899 matrixof finite order congruent to 119868119899(modI)

(i) If 120594 = 119901 gt 0 then 119892119901119895

= 119868119899 for some integer 119895 If 120594 = 0then 119868 contains a prime number 119901 and 119892

119901119894

= 119868119899 forsome integer 119894 In particular a finite group of matricescongruent to 119868119899(modI) is a 119901-group

(ii) Let 120594 = 0 and let I = p be a prime idealhaving ramification index 119890 with respect to 119901 let 119892 equiv

119868119899(modp119903) and let

120582119901119894minus1

(119901 minus 1) ⩽119890

119903lt 119901119894(119901 minus 1) 119894 ⩾ 0 120582 = min 1 119894

(3)

Then 119892119901119894

= 119868119899 in particular any finite group of matricescongruent to 119868119899(modp119905) is trivial if 119890 lt 119903(119901 minus 1)

See [17 Lemma 1] for the proof of Proposition 3The following corollary can be immediately obtained

from Proposition 3

Corollary 4 Let 119870 be a number field of degree 119889 = [119870

Q] with the maximal order 119874119870 The kernel of reduction of119866119871119899(119874119870)modulo an idealI of119874119870 containing a prime number119901 has no torsion if the norm119873119870Q(I) gt 119901

119889(119901minus1)

Remark 5 Earlier Burgisser obtained a similar result for119873119870Q(I) gt 2

119889 see [18 Lemma 31]

Propositions 3 and 6 below can be used for estimating theorders of finite subgroups of GL119899(O119870) using the reductionmodulo some prime ideal p sub 119874119870 It is also possible to

determine the structure of a 119901-subgroup of GL119899(119874119870) havingthe maximal possible order with some modifications in thecase 119901 = 2 The theorems describing the maximal 119901-subgroups of GL119899(119870) over fields can be found in [19] inparticular it is proven that there is only one conjugacy classof maximal 119901-subgroups of GL119899(119870) for 119901 gt 2 see also [9 20]However the equivalence of subgroups in GL119899(119874119870) over 119874119870is a more subtle question See [21 chapter 3] [22 23] for thestructure of finite linear groups (including the groups of smallorders) See [15] for more details proofs and applications todetermining torsion elements in the reduced projective classgroup

As a corollary of Theorem 1 we can obtain the followingproposition

Proposition 6 Let 119870Q119901 be a finite extension and let 120577119901 isin

119874119870 Let 119901 = p119890 and let 119890 = 119901 minus 1 Let 119866 be a finite subgroupof 119866119871119899(119874119870) and 119892 equiv 119868119899(modp) for all 119892 isin 119866 Then 119866 isconjugate in119866119871119899(119874119870) to an abelian group of diagonal matricesof exponent 119901

Proof of Proposition 6 Let us prove that 119866 is abelian of expo-nent 119901 Let 120587 be a prime element of 119874119870 Let 1198921 = 119868119899 + 12058711986111198922 = 119868119899 + 1205871198612 for some 1198921 1198922 isin 119866 Then 119892

minus1

119894equiv 119868119899 minus

120587119861119894(mod1205872) 119894 = 1 2 and ℎ = 11989211198922119892minus1

1119892minus1

2equiv 119868119899(mod1205872)

It follows from Proposition 2 that ℎ = 119868119899 and the sameproposition shows that 119892119901 = 119868119899 for any 119892 isin 119866 First of all 119866is conjugate over 119874119870 to a group of triangular matrices since119866 is abelian and 119874119870 is a local ring see [6 Theorem 739] andthe remarks in [6 on page 493] Following Theorem 1 let usprove that every 119892 isin 119866 is diagonalizable We can describeexplicitly the matrix119872 such that

119872minus1119892119872 = diag (1205821 1205822 120582119899) (4)

is a diagonalmatrix for a triangularmatrix 119892 of order119901whichis congruent to 119868119899(modp) Indeed let 119892 isin 119866 and

119892 =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577(1)1198681199051

1198751

2 119875

1

119896

0 120577(2)1198681199052

1198752

119896

d

0 sdot sdot sdot 120577(119896)119868119905119896

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(5)

and let

119878 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198681199051

0 1198601

0 1198681199052

1198602

d

0 sdot sdot sdot 119868119905119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(6)

for 1199051 + 1199052 + sdot sdot sdot + 119905119896 = 119899 and 1199051 le 1199052 le sdot sdot sdot le 119905119896 120577(119894) 119894 =

1 2 119896 are appropriate 119901-roots of 1 We consider

119878minus1119892119878 =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577(1)1198681199051

lowast 1198721

119896

0 120577(2)1198681199052

1198722

119896

d

0 sdot sdot sdot 120577(119896)119868119905119896

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(7)

ISRN Algebra 3

and we find the system of conditions for providing 119872119894

119896=

0119905119894119905119896

the zero 119905119894 times 119905119896 matrix We have the following systemof conditions

120577(1) (1 minus 120577(119896)120577minus1

(1))1198601 + 119875

1

21198602 + sdot sdot sdot + 119875

1

119896minus1119860119896minus1 + 119875

1

119896= 01199051119905119896

sdot sdot sdot

120577(119896minus2)119860119896minus2 (1 minus 120577(119896)120577minus1

(119896minus2)) + 119875119896minus2

119896minus1119860119896minus1 + 119875

119896minus2

119896= 0119905119896minus2119905119896


(119896minus1)) + 119875119896minus1

119896= 0119905119896minus1119905119896

(8)

The condition 119892 equiv 119868119899(modp) implies that 119875119895

119894equiv

0119905119895119905119894

(modp) and we can find 119860 119894 1 le 119894 le 119896 minus 1 sequentiallyusing the results of the previous steps

119860119896minus1 = minus119875119896minus1

119896

120577(119896minus1) (1 minus 120577(119896)120577minus1

(119896minus1))

119860119896minus2 = minus

(119875119896minus2

119896+ 119875119896minus2

119896minus1119860119896minus1)


(119896minus2))

119860119896minus3 = minus

(119875119896minus3

119896+ 119875119896minus3

119896minus1119860119896minus1 + 119875

119896minus3

119896minus2119860119896minus2)


(119896minus3))

(9)

and so on Now using the induction on the degree 119899 we canfind a matrix 119872 that transforms 119892 to a diagonal form asrequired

The condition 119892 equiv 119868119899(modp) of Proposition 6 impliesthat the condition (i) of Theorem 1 holds true Since 119866 is anabelian group of exponent 119901 this allows us to prove our claimover the ring 119874119870

Remark 7 Using the same argument for an algebraic numberfields119870Qwe can prove a similar result Let119874 be a Dedekindring in 119870 and let 120577119901 isin 119874 Let 119901 = p119890 119890 = 119901 minus 1 Let 119866 be afinite subgroup of GL119899(119874) and 119892 equiv 119868119899(modp) for all 119892 isin 119866Then119866 is conjugate inGL119899(119874) to an abelian group of diagonalmatrices of exponent 119901

In this situation we can use the statement (8120) in [CR]for proving the above result for the given Dedekind ring 119874

(compare also the proof of (8120) and (7527) in [6])However in Proposition 6 the ramification index 119890 = 119901 minus

1 Below there are two examples giving constructions of localfield extensions 119870Q119901 and finite subgroups 119866 sub GL119899(119874119870)which are contained in the kernel of reduction of GL119899(119874119870)modulo ideals having ramification indices 119890 gt 119901 minus 1

Example 8 For the following finite extension 119870Q119901 of localfields obtained via adjoining torsion points of elliptic curveslet 119874119870 be the ring of integers of 119870 with the maximal idealp Consider an elliptic curve 119864 over Z119901 with supersingulargood reduction (see [24 Section 111]) Let 119870Q119901 be the field

extension obtained by adjoining 119901-torsion points of 119864 thenthe formal group associated with 119864 has a height of 2 its Hopfalgebra 119874119860 is a free module of rank 119901

2 over Z119901 and for thekernel 119864119901 of multiplication by 119901 |119864119901| = 119901

2 (see [25 13 andSection 2]) Note that for some 119864 the ramification index 119890 =

119890(119870Q119901) = 1199012minus 1 ([24 page 275 Proposition 12])

We can consider the group 119866 of 119901-torsion points as Z119901-algebra homomorphisms from the Hopf algebra 119874119860 to theZ119901-algebra 119874119870 then 119866 = HomZ

119901

(119874119860 119874119870) and the algebra

119874119860 is isomorphic toZ119901[119883](1198881119883+ 11988821198832+ sdot sdot sdot + 119883

1199012

) see [25Section 2] and [26] So there is a representation V 119866 rarr

GL1199012(119874119870) and since 119864 is supersingular the image of V iscontained in the kernel of reduction modulo p

The following example shows that the kernel of reductionof GL119899(119874119870) modulo a prime divisor of 119901 may contain 119901-groups of any prescribed nilpotency class 119897 gt 1 for extensions119870Q119901 with large ramification these groups are not abelianand they are not diagonalizable in GL119899(119874119870)

Example 9 Let us consider the following 119901-group of nilpo-tency class 119897 determined by generators 119886 1198871 119887119897 and rela-tions 119887119901

119894= 1 119887119894119887119895 = 119887119895119887119894 119894 = 1 2 119897 1198861198871 = 1198871119886 119887119894minus1 =

119887119894119886119887minus1

i 119886minus1 119894 = 1 2 119897 119886119899 = 1 where 119899 = 119901

119905ge 119897 gt 119901

119905minus1 and119905 is a suitable positive integer Let119867 be the abelian subgroupof119866 generated by 1198871 119887119897 and let 120594 denotes the character of119867 given on the generators as follows 120594(1198871) = 120577119901 minus119886 primitive119901-root of 1 120594(119887119894) = 1 119894 = 2 119897 The character 120594 togetherwith the decomposition of 119866 into cosets with respect to 119867119866 = 1 sdot 119867 + 119886 sdot 119867 + sdot sdot sdot + 119886

119899minus1sdot 119867 gives rise to an induced

representation 119877 = Ind 119866119867120594 of 119866 For the 119899 times 119899 matrices 119890119894119895

having precisely one nonzero entry in the position (119894 119895) equalto 1 we can define a 119899 times 119899 matrix using the binomial coef-ficients ( 119899minus119895

119894minus119895)

119862 = Σ119899 ge 119894 ge 119895 ge 1(minus1)119894minus119895

(119899 minus 119895

119894 minus 119895) 119890119894119895 (10)

Theorem 10 LetQ119901(120577119901infin) denote the extension ofQ119901 obtainedby adjoining all roots 120577119901119894 119894 = 1 2 3 of 119901-primary orders of1 let 120587 be the uniformizing element of a finite extension 119870Q119901

such that 119870 sub Q119901(120577119901infin) and let 119863 = diag(1 120587 1205872 120587119899minus1)Then for 119892 isin 119866 the representation 119877120587(119892) = 119863

minus1119862minus1119877(119892)119862119863

of 119866 is a faithful absolute irreducible representation in119866119871119899(119874119870) by matrices congruent to 119868119899(mod120587) Moreover suchrepresentations are pairwise nonequivalent over 119874Q

119901(120577119901infin ) and

for the lower central series 119866 = 119866119897 sup 119866119897minus1 sup sdot sdot sdot sup 1198660 = 119868119899

of 119866 all elements of 119877120587(119866119897minus119894+1) are congruent to 119868119899(mod120587119894119908) ifthe elements of 119877120587(119866) are congruent to 119868119899(mod120587119908)

For the proof of Theorem 10 (which is constructive) see[17 27] Remark that the construction of Theorem 10 can berealized also over the integers of cyclotomic subextensions119870 sub Q(120577119901infin) = ⋃

infin

119894=1Q(120577119901119894) ofQ and other global fields

Acknowledgment

The author is grateful to the referees for useful suggestions

4 ISRN Algebra

References

[1] W C Brown Matrices Over Commutative Rings vol 169 ofMonographs and Textbooks in Pure and Applied MathematicsMarcel Dekker New York NY USA 1993

[2] D Laksov ldquoDiagonalization of matrices over ringsrdquo Journal ofAlgebra vol 376 pp 123ndash138 2013

[3] H Minkowski ldquoUberber den arithmetischen Begriff derAquivalenz und uber die endlichen Gruppen linearer ganz-zahliger Substitutionenrdquo Journal fur Die Reine und AngewandteMathematik vol 100 pp 449ndash458 1887

[4] H Minkowski ldquoZur Theorie der positiven quadratischen For-menrdquo Journal fur Die Reine und Angewandte Mathematik vol101 no 3 pp 196ndash202 1887

[5] H Minkowski Geometrie der Zahlen Teubner Berlin Ger-many 1910

[6] C W Curtis and I Reiner Representation Theory of FiniteGroups and Associative Algebras Interscience Publishers NewYork NY USA 1962

[7] C W Curtis and I Reiner Methods of Representation YheoryVol I John Wiley amp Sons New York NY USA 1981

[8] C W Curtis and I Reiner Methods of Representation TheoryVol II vol 2 John Wiley amp Sons New York NY USA Withapplications to finite groups and orders 1987

[9] W Feit ldquoFinite linear groups and theorems of Minkowski andSchurrdquo Proceedings of the American Mathematical Society vol125 no 5 pp 1259ndash1262 1997

[10] H-J Bartels ldquoZur Galoiskohomologie definiter arithmetischerGruppenrdquo Journal fur die Reine und Angewandte Mathematikvol 298 pp 89ndash97 1978

[11] J Rohlfs ldquoArithmetisch definierte Gruppen mit Galoisopera-tionrdquo InventionesMathematicae vol 48 no 2 pp 185ndash205 1978

[12] H-J Bartels and D A Malinin ldquoFinite Galois stable subgroupsof 119866119871119899rdquo in Noncommutative Algebra and Geometry C deConcini F van Oystaeyen N Vavilov and A Yakovlev Edsvol 243 of Lecture Notes in Pure and Applied Mathematics pp1ndash22 2006

[13] H-J Bartels and D A Malinin ldquoOn finite Galois stablesubgroups of 119866119871119899 in some relative extensions of number fieldsrdquoJournal of Algebra and its Applications vol 8 no 4 pp 493ndash5032009

[14] D A Malinin ldquoGalois stability for integral representations offinite groupsrdquo Rossiıskaya Akademiya Nauk Algebra i Analizvol 12 no 3 pp 106ndash145 2000 (Russian)

[15] B Burgisser ldquoOn the projective class group of arithmeticgroupsrdquo Mathematische Zeitschrift vol 184 no 3 pp 339ndash3571983

[16] P Roquette ldquoRealisierung von Darstellungen endlicher nilpo-tenter Gruppenrdquo Archiv der Mathematik vol 9 pp 241ndash2501958

[17] D A Malinin ldquoIntegral representations of 119901-groups over localfieldsrdquo Doklady Akademii Nauk SSSR vol 309 no 5 pp 1060ndash1063 1989 (Russian)

[18] B Burgisser ldquoFinite 119901-periodic quotients of general lineargroupsrdquo Mathematische Annalen vol 256 no 1 pp 121ndash1321981

[19] C R Leedham-Green andW Plesken ldquoSome remarks on Sylowsubgroups of general linear groupsrdquoMathematische Zeitschriftvol 191 no 4 pp 529ndash535 1986

[20] J-P Serre ldquoBounds for the orders of the finite subgroups of119866(119896)rdquo inGroupRepresentationTheory pp 405ndash450 EPFLPressLausanne Switzerland 2007

[21] A E Zalesskii ldquoLinear groupsrdquo Russian Mathematical Surveysvol 36 pp 63ndash128 1981

[22] A E Zalesskiı ldquoLinear groupsrdquo inAlgebra Topology Geometryvol 21 of Itogi Nauki i Tekhniki pp 135ndash182 Moscow Russia1983

[23] P H Tiep and A E Zalesskii ldquoSome aspects of finite lineargroups a surveyrdquo Journal of Mathematical Sciences vol 100 no1 pp 1893ndash1914 2000

[24] J-P Serre ldquoProprietes galoisiennes des points drsquoordre fini descourbes elliptiquesrdquo InventionesMathematicae vol 15 no 4 pp259ndash331 1972

[25] F Destrempes ldquoDeformations of Galois representations the flatcaserdquo in Seminar on Fermatrsquos Last Theorem vol 17 pp 209ndash231American Mathematical Society Providence RI USA 1995

[26] V A Kolyvagin ldquoFormal groups and the norm residue symbolrdquoIzvestiya Akademii Nauk SSSR vol 43 no 5 Article ID10541120 1979

[27] D A Malinin ldquoIntegral representations of 119901-groups of a givenclass of nilpotency over local fieldsrdquo Rossiıskaya AkademiyaNauk Algebra i Analiz vol 10 no 1 pp 58ndash67 1998 (Russian)

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Stochastic AnalysisInternational Journal of

2 ISRN Algebra

Proof of Theorem 1 Let us denote by 119863 the subgroup of allscalar matrices in 119866 and let 119885 be the centre of 119866

If119863 =119885 we use Proposition 2 and induction on 119899Let119863 = 119885 and let the exponent of the group119885 be equal to

119905 Let1198920 isin 119866 be any element not contained in the centre119885 of119866such that the image of 1198920 in the factor group119866119885 is containedin the centre of 119866119885 Then we consider the homomorphism

120595 1198661015840997888rarr GL119899119905 (119860) (2)

given by 120595(119892) = 119892otimes119905

The kernel of 120595 is the set 119863 of all scalarmatrices contained in 119866 This kernel is not trivial since 119866 isnilpotent The image 120595(119866) is isomorphic to the factor group119866119863 Let 1198920 = diag(1198891 119889119899) Then 120595(1198920) is a nonscalardiagonal matrix in the centre of 120595(119866) 1198920 is also not a scalarmatrix and for any 119892 isin 119866 we have 1198921198920 = 1198920119892120577 for someroot of 1120577 = 120577(119892) If for the matrix 119892 = [119892119899119896]119899119896 the elements119892119894119895 and 119892119895119894 are not zero we obtain 119889119895 = 119889119894120577 and 119889119894 = 119889119895120577which implies immediately that 120577 = 1 if 119894 = 119895 (as in case(i) of Theorem 1) In case (ii) of Theorem 1 if 119894 = 119895 we obtain1205772= 1 and 120577 = plusmn1 but 120577 = minus1 is impossible in the virtue of the

condition thatminus119868119899 is not contained in119866 Hencewe have 120577 = 1and 1198920 is contained in 119885 the centre of 119866 This contradictioncompletes the proof of Theorem 1

Proposition 3 Let I be an ideal of a Dedekind ring 119878 ofcharacteristic 120594 let 0 =I = 119878 and let 119892 be a 119899 times 119899 matrixof finite order congruent to 119868119899(modI)

(i) If 120594 = 119901 gt 0 then 119892119901119895

= 119868119899 for some integer 119895 If 120594 = 0then 119868 contains a prime number 119901 and 119892

119901119894

= 119868119899 forsome integer 119894 In particular a finite group of matricescongruent to 119868119899(modI) is a 119901-group

(ii) Let 120594 = 0 and let I = p be a prime idealhaving ramification index 119890 with respect to 119901 let 119892 equiv

119868119899(modp119903) and let

120582119901119894minus1

(119901 minus 1) ⩽119890

119903lt 119901119894(119901 minus 1) 119894 ⩾ 0 120582 = min 1 119894

(3)

Then 119892119901119894

= 119868119899 in particular any finite group of matricescongruent to 119868119899(modp119905) is trivial if 119890 lt 119903(119901 minus 1)

See [17 Lemma 1] for the proof of Proposition 3The following corollary can be immediately obtained

from Proposition 3

Corollary 4 Let 119870 be a number field of degree 119889 = [119870

Q] with the maximal order 119874119870 The kernel of reduction of119866119871119899(119874119870)modulo an idealI of119874119870 containing a prime number119901 has no torsion if the norm119873119870Q(I) gt 119901

119889(119901minus1)

Remark 5 Earlier Burgisser obtained a similar result for119873119870Q(I) gt 2

119889 see [18 Lemma 31]

Propositions 3 and 6 below can be used for estimating theorders of finite subgroups of GL119899(O119870) using the reductionmodulo some prime ideal p sub 119874119870 It is also possible to

determine the structure of a 119901-subgroup of GL119899(119874119870) havingthe maximal possible order with some modifications in thecase 119901 = 2 The theorems describing the maximal 119901-subgroups of GL119899(119870) over fields can be found in [19] inparticular it is proven that there is only one conjugacy classof maximal 119901-subgroups of GL119899(119870) for 119901 gt 2 see also [9 20]However the equivalence of subgroups in GL119899(119874119870) over 119874119870is a more subtle question See [21 chapter 3] [22 23] for thestructure of finite linear groups (including the groups of smallorders) See [15] for more details proofs and applications todetermining torsion elements in the reduced projective classgroup

As a corollary of Theorem 1 we can obtain the followingproposition

Proposition 6 Let 119870Q119901 be a finite extension and let 120577119901 isin

119874119870 Let 119901 = p119890 and let 119890 = 119901 minus 1 Let 119866 be a finite subgroupof 119866119871119899(119874119870) and 119892 equiv 119868119899(modp) for all 119892 isin 119866 Then 119866 isconjugate in119866119871119899(119874119870) to an abelian group of diagonal matricesof exponent 119901

Proof of Proposition 6 Let us prove that 119866 is abelian of expo-nent 119901 Let 120587 be a prime element of 119874119870 Let 1198921 = 119868119899 + 12058711986111198922 = 119868119899 + 1205871198612 for some 1198921 1198922 isin 119866 Then 119892

minus1

119894equiv 119868119899 minus

120587119861119894(mod1205872) 119894 = 1 2 and ℎ = 11989211198922119892minus1

1119892minus1

2equiv 119868119899(mod1205872)

It follows from Proposition 2 that ℎ = 119868119899 and the sameproposition shows that 119892119901 = 119868119899 for any 119892 isin 119866 First of all 119866is conjugate over 119874119870 to a group of triangular matrices since119866 is abelian and 119874119870 is a local ring see [6 Theorem 739] andthe remarks in [6 on page 493] Following Theorem 1 let usprove that every 119892 isin 119866 is diagonalizable We can describeexplicitly the matrix119872 such that

119872minus1119892119872 = diag (1205821 1205822 120582119899) (4)

is a diagonalmatrix for a triangularmatrix 119892 of order119901whichis congruent to 119868119899(modp) Indeed let 119892 isin 119866 and

119892 =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577(1)1198681199051

1198751

2 119875

1

119896

0 120577(2)1198681199052

1198752

119896

d

0 sdot sdot sdot 120577(119896)119868119905119896

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(5)

and let

119878 =

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198681199051

0 1198601

0 1198681199052

1198602

d

0 sdot sdot sdot 119868119905119896

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(6)

for 1199051 + 1199052 + sdot sdot sdot + 119905119896 = 119899 and 1199051 le 1199052 le sdot sdot sdot le 119905119896 120577(119894) 119894 =

1 2 119896 are appropriate 119901-roots of 1 We consider

119878minus1119892119878 =

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120577(1)1198681199051

lowast 1198721

119896

0 120577(2)1198681199052

1198722

119896

d

0 sdot sdot sdot 120577(119896)119868119905119896

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(7)

ISRN Algebra 3


119896=

0119905119894119905119896


120577(1) (1 minus 120577(119896)120577minus1

(1))1198601 + 119875

1

21198602 + sdot sdot sdot + 119875

1

119896minus1119860119896minus1 + 119875

1

119896= 01199051119905119896

sdot sdot sdot


(119896minus2)) + 119875119896minus2

119896minus1119860119896minus1 + 119875

119896minus2

119896= 0119905119896minus2119905119896


(119896minus1)) + 119875119896minus1

119896= 0119905119896minus1119905119896

(8)


119894equiv

0119905119895119905119894



119896


(119896minus1))

119860119896minus2 = minus

(119875119896minus2

119896+ 119875119896minus2

119896minus1119860119896minus1)


(119896minus2))

119860119896minus3 = minus

(119875119896minus3

119896+ 119875119896minus3

119896minus1119860119896minus1 + 119875

119896minus3

119896minus2119860119896minus2)


(119896minus3))

(9)













119901

(119874119860 119874119870) and the algebra


1199012





119894= 1 119887119894119887119895 = 119887119895119887119894 119894 = 1 2 119897 1198861198871 = 1198871119886 119887119894minus1 =

119887119894119886119887minus1

i 119886minus1 119894 = 1 2 119897 119886119899 = 1 where 119899 = 119901

119905ge 119897 gt 119901





119894minus119895)


(119899 minus 119895

119894 minus 119895) 119890119894119895 (10)



minus1119862minus1119877(119892)119862119863


119901(120577119901infin ) and




infin


Acknowledgment


4 ISRN Algebra

References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









ISRN Algebra 3


119896=

0119905119894119905119896


120577(1) (1 minus 120577(119896)120577minus1

(1))1198601 + 119875

1

21198602 + sdot sdot sdot + 119875

1

119896minus1119860119896minus1 + 119875

1

119896= 01199051119905119896

sdot sdot sdot


(119896minus2)) + 119875119896minus2

119896minus1119860119896minus1 + 119875

119896minus2

119896= 0119905119896minus2119905119896


(119896minus1)) + 119875119896minus1

119896= 0119905119896minus1119905119896

(8)


119894equiv

0119905119895119905119894



119896


(119896minus1))

119860119896minus2 = minus

(119875119896minus2

119896+ 119875119896minus2

119896minus1119860119896minus1)


(119896minus2))

119860119896minus3 = minus

(119875119896minus3

119896+ 119875119896minus3

119896minus1119860119896minus1 + 119875

119896minus3

119896minus2119860119896minus2)


(119896minus3))

(9)













119901

(119874119860 119874119870) and the algebra


1199012





119894= 1 119887119894119887119895 = 119887119895119887119894 119894 = 1 2 119897 1198861198871 = 1198871119886 119887119894minus1 =

119887119894119886119887minus1

i 119886minus1 119894 = 1 2 119897 119886119899 = 1 where 119899 = 119901

119905ge 119897 gt 119901





119894minus119895)


(119899 minus 119895

119894 minus 119895) 119890119894119895 (10)



minus1119862minus1119877(119892)119862119863


119901(120577119901infin ) and




infin


Acknowledgment


4 ISRN Algebra

References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 ISRN Algebra

References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article On Finite Nilpotent Matrix Groups over ...ISRN Algebra References [] W. C. Brown,...

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