On minimal artinian modules and minimal artinian linear groups

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PII. S0161171201007165 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON MINIMAL ARTINIAN MODULES AND MINIMAL

ARTINIAN LINEAR GROUPS

LEONID A. KURDACHENKO and IGOR YA. SUBBOTIN

(Received 19 March 2001)

Abstract.The paper is devoted to the study of some important types of minimal artinian linear groups. The authors prove that in such classes of groups as hypercentral groups (so also, nilpotent and abelian groups) andF C-groups, minimal artinian linear groups have precisely the same structure as the corresponding irreducible linear groups.

2000 Mathematics Subject Classiﬁcation. 20E36, 20F28.

LetF be a field,Aa vector space overF. The group GL(F , A)of all automorphisms ofAand its distinct subgroups are the oldest subjects of investigation in Group The-ory. For the case when Ahas a finite dimension overF, every element of GL(F , A) defines some nonsingularn×n-matrix overF, wheren=dimFA. Thus, for the

finite-dimensional case, the theory of linear groups is exactly the theory of matrix groups. That is why the theory of finite-dimensional linear groups is one of the best developed in algebra. However, for the case when dimFAis infinite, the situation is totally

dif-ferent. The study of this case always requires some essential additional restrictions. Thus, the transition from the study of finite groups to the study of infinite groups generated the finiteness conditions. It is natural to apply these finiteness conditions to the study of infinite-dimensional linear groups. The study of finitary linear groups (the linear analogies ofF C-groups) shows the effectiveness of such approach (cf. a survey of Phillips [6]).

The groups having a finite composition series were one of the first generalization of the finite groups. Let G ≤GL(F , A), then we can consider A as an F G-module. We say that A has a finite composition length if A has a finite series0 = B0≤ B1≤ ··· ≤Bn=AofF G-submodules, every factor of which is a simpleF G-module.

We can consider G/CG(Bi+1/Bi) as an irreducible linear group, 0≤ i≤ n−1. Let

T =0≤i≤n−1CG(Bi+1/Bi); then G/T ≤X0≤i≤n−1G/CG(Bi+1/Bi),and T is a nilpotent

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The minimal and the maximal conditions were the very next classical finiteness conditions that have appeared in algebra. Note that everyF G-module of finite com-position length is artinian (i.e., it satisfies the minimal condition onF G-submodules) and noetherian (i.e., it satisfies the maximal condition onF G-submodules).

LetRbe a ring,Aan artinianR-module. Put

Sicl(A)=B|Bis anR-submodule ofAand has no ﬁnite composition series. (1)

IfAhas no ﬁnite composition series, thenSicl(A)≠∅. SinceAis artinian,Sicl(A) has a minimal elementM. Thus, ifUis a properR-submodule ofM,thenUhas a ﬁnite composition length.

AnR-moduleM is said to be a minimal artinian, ifM has no ﬁnite composition series, but each of its proper submodule has a ﬁnite composition length.

Thus every artinian module includes a minimal artinian submodule. On the other hand, the structure of artinian modules depends on the structure of its minimal ar-tinian submodules, therefore, the study of minimal arar-tinian modules is one of the important steps for the study of artinian modules.

Let againF be a field,Aa vector space overF,G≤GL(F , A). We want to consider the situation whenAis a minimal artinianF G-module. This consideration will lead us to the fact that the groupGis lying in the classXsuch that all irreducibleX-groups have been described. So we may set that if an F G-moduleB has finite composition series, then the structure ofGis defined.

LetF be a ﬁeld,Aa vector space overF,G≤GL(A). A groupGis called a minimal artinian if the following conditions hold:

(MA1)Ahas no ﬁnite composition series;

(MA2) ifBis a properF G-submodule ofA,thenBhas a ﬁnite composition length. The study of minimal artinian F G-modules (as any F G-module) consists of two parts: the study of internal structure of the module and the study of the group G/CG(A). The last group is imbedded in GL(F , A), that is, it is a linear minimal

ar-tinian group. Our paper is devoted to the study of some important types of minimal artinian linear groups. The main results of this paper show that in such classes of groups as hypercentral groups (so also, nilpotent and abelian groups) orF C-groups the minimal artinian linear groups have precisely the same structure as the corre-sponding irreducible linear groups have.

Now we mention some needed results on hypercentral irreducible groups. The ir-reducibleZG-modules have been studied in [5]. These results can be extended almost without changes on the case of irreducible subgroups of GL(F , A), whereAis a vector space over a ﬁeldF.

Lemma1. LetF be a ﬁeld,Ga group,Aa simpleF G-module,I=Ann_{F G}(A). IfC/I

is a center ofF G/I, thenC/I is an integral domain. In particular, the periodic part of

ζ(G/CG(A))is a locally cyclicp-subgroup wherep=charF.

As usual, 0denotes the set of all primes.

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We note that every refinement of each of these series has only r infinite cyclic factors. Since every two subnormal series have the isomorphic refinements, 0-rank is independent of the choice of the subnormal series.

Note also that ifGis a locally nilpotent group of ﬁnite 0-rank, then the factor-group G/t(G)by the periodic partt(G)has a ﬁnite special rank.

Lemma₂. _Let_G_{be a hypercentral group of finite}₀_-rank,_F _{a locally finite field,}_A_a

simpleF G-module. Thenζ(G/CG(A))is periodic.

This lemma follows from [4, Theorem 2].

Lemma3. LetF be a ﬁeld,p=charF,Gan abelian group of ﬁnite0-rank.

(1)If the ﬁeldF is locally ﬁnite, andGis a locally cyclicp-group, then there exists a

simpleF G-moduleAsuch thatCG(A)= 1.

(2)IfF is not locally ﬁnite, andt(G)is a locally cyclicp-group, then there exists a

simpleF G-moduleAsuch thatCG(A)= 1.

This construction is contained in [2].

Lemma₄. _Let_F_{be a ﬁeld,}_p=_char_F_,_G_{an abelian group of inﬁnite}₀_{-rank. If}_t(G)_is

a locally cyclicp-group, then there exists a simpleF G-moduleAsuch thatCG(A)= 1.

This assertion has been proved in [9] for the case of finite field, however it is valid also for an arbitrary field.

Lemma ₅. _Let _F _{be a ﬁeld,} _p =_char_F_, _G _{a hypercentral group of ﬁnite} ₀_-rank, C=ζ(G),T=t(C).

(1)If the ﬁeldF is locally ﬁnite, andC =T is a locally cyclicp-group, then there

exists a simpleF G-moduleAsuch thatCG(A)= 1.

(2)IfFis not locally ﬁnite andT is a locally cyclicp-group, then there exists a simple

F G-moduleAsuch thatCG(A)= 1.

Lemma₆. _Let_F _{be a ﬁeld,}_p=_char_F_, _G_{a hypercentral group of inﬁnite} ₀_-rank,

C=ζ(G), T =t(C). IfT is a locally cyclicp-group, then there exists a simpleF G

-moduleAsuch thatCG(A)= 1.

The proof of both these assertions is similar to the proof of the respective results of [5].

Lemma₇. _Let_R_{be a ring,}_A_{a minimal artinian}_R_{-module. Then}_A_{does not}

decom-pose into a direct sum of two properR-submodules.

The lemma is obvious.

IfAis anR-module, then let SocR(A)denotes the sum of all minimalR-submodules

wheneverAincludes such submodules, and SocR(A)= 0otherwise.

Clearly, SocR(A)is a direct sum of some minimalR-submodules (if it is nonzero). If

Ais an artinianR-module, then SocR(A)≠0and SocR(A)is a direct sum of ﬁnitely

many minimalR-submodules. So we come to the following lemma.

Lemma₈. _Let_R_{be a ring,}_A_{a minimal artinian}_R_{-module. Then}_Soc_R_(A)_{is a nonzero}

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Lemma₉. _Let_F _{be a ﬁeld,}_G_{a group,}_H _{a normal subgroup having a ﬁnite index}

inG,AanF G-module. IfAhas ﬁnite composition length as anF G-module, thenAhas

ﬁnite composition length as anF H-module.

Proof. _Let

0 =B0≤B1≤ ··· ≤Bn=A (2)

be a series ofF G-submodules with simpleF G-factors. ThenBi+1/Biis a direct sum of

ﬁnitely many simpleF H-submodules [12], 0≤i≤n−1. ThusAhas a ﬁnite series of F H-submodules with simple factors.

Proposition₁₀. _Let_F _{be a ﬁeld,}_G_{a group,}_A_{a minimal artinian}_{F G}_{-module such}

thatCG(A)= 1,Ha normal subgroup having inGﬁnite index,Xa transversal toH

inG. Then

(1) Aincludes a minimal artinianF H-submoduleB;

(2) A=x∈XBx;

(3) _x∈X x−1_C

H(B)x= 1, in particular,H≤Xx∈XH/(x−1CH(B)x).

Proof. _{By Wilson’s theorem [}₁₁_]_A_{is an artinian}_{F H-module. Since}_A_{has no ﬁnite} composition series as F G-module, then A has no ﬁnite composition series as F H-module byLemma 9. Let

S= {U|Uis anF H-submodule ofAand has no ﬁnite composition series}. (3)

SinceA∈S,S≠∅. ThenShas a minimal elementB. This means thatBis minimal artinianF H-submodule. The sumC=x∈XBxis anF G-submodule. If we suppose

that C is a proper F G-submodule ofA, then it has a ﬁnite composition length. By

Lemma 9it has also a ﬁnite composition length as anF H-module, which contradicts the choice ofB. This contradiction proves the equalityA=x∈XBx. SinceCH(Bx)=

x−1_C

H(B)x, then it follows thatx∈Xx−1CH(B)x≤CH(A)= 1. By Remak’s theorem,

H≤X_x∈XH/(x−1_C

H(B)x).

Lemma₁₁. _Let_F _{be a ﬁeld,}_G_{a group,}_A_{a minimal artinian}_{F G}_{-module such that}

CG(A)= 1. If1≠x∈ζ(G),thenA=A(x−1).

Proof. The mappingϕ:a→a(x−1),a∈A, is an F G-endomorphism ofA. In particular, Imϕ=A(x−1)and Kerϕ=CA(x)are theF G-submodules ofA. Since

x∈CG(A), thenCA(x)≠A. ByA(x−1)A/CA(x), we obtain thatA(x−1)has no

ﬁnite composition length. It follows thatA(x−1)=A.

Corollary ₁₂. _Let_F _{be a ﬁeld,}_A _{a vector space over} _F_, _G _{a minimal artinian}

subgroup ofGL(F , A). Suppose thatGis hypercentral. IfcharF =p >0, thenGdoes

not containp-elements.

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LetGbe a group. Put

F C(G)=x∈G|xG=g−1xg|g∈Gis ﬁnite. (4)

That is,F C(G)is a characteristic subgroup ofG. This subgroup is called theF C-center ofG.

Furthermore, the setT of all elements of ﬁnite order is a (characteristic) subgroup ofF C(G)andF C(G)/T is an abelian torsion-free group (cf. [7, Theorem 4.32]).

Let G be a group, π a set of primes. Denote byOπ(G) the maximal normal

π-subgroup ofG. In particular, ifpis prime, thenOp(G)denotes the maximal normal

p-subgroup ofG, andOp(G)denotes the maximal periodic subgroup, which does not

contain thep-elements.

Corollary 13. LetF be a ﬁeld,A a vector space over F, G a minimal artinian

subgroup ofGL(F , A). IfcharF=p >0, thenOp(F C(G))= 1.

Proof. _{Suppose the contrary, let 1}≠_y∈_O_p_{(F C(G)). Put}_Y= _yG_{. By Ditsmann’s} lemma (cf. [7, Corollary 2 to Lemma 2.14]),Y is a ﬁnite normal subgroup ofG. Since Y is a ﬁnitep-subgroup, ζ(Y )=Z≠1. LetH=CG(Z), thenH is a normal

sub-group of ﬁnite index, and Z ≤ ζ(H). By Proposition 10 A includes a minimal ar-tinian F H-submoduleB. Since the additive group of B is an elementary abelian p-group, the natural semidirect productB zis a nilpotent group for each element z∈Z(cf. [8, Lemma 6.34]). Therefore[Bz, Bz]=B(z−1)≠B.Corollary 12yields that z ∈CG(B). It is valid for every element z∈ Z, thereforeZ ≤ CG(B). In turn

Z=x−1_Zx_≤_x−1_C

G(B)x=CG(Bx)for an arbitrary elementx∈G. Since it is true

for every element x ∈ G, Z ≤ x∈GCG(Bx) = CG(A), because A =

x∈XBx. But

CG(A)= 1. This contradiction proves thatOp(F C(G))= 1.

Corollary 14. LetF be a ﬁeld,A a vector space over F, G a minimal artinian

subgroup ofGL(F , A). IfcharF=p >0, then the locally soluble radical ofF C(G)has

nop-elements.

Lemma15. LetF be a ﬁeld,charF =p,Aa vector space overF,Ga minimal

ar-tinian subgroup ofGL(F , A). IfHis a nonidentity ﬁnite normalp-subgroup ofG, then

SocF H(A)=A.

Proof. For every element 0≠a∈A, anF H-submoduleaF His ﬁnite-dimensional. In particular, it includes a simpleF H-submodule. This means that SocF H(A)≠0. By

Maschke’s theorem (cf. [10, Theorem 1.5]), SocF H(A)=A.

IfRis a ring, Ga group, thenωRGdenotes the augmentation ideal of the group ringRG.

Corollary16. LetF be a ﬁeld,charF=p,Aa vector space overF,Ga minimal

artinian subgroup of GL(F , A). IfH is a nonidentity ﬁnite normalp-subgroup of G,

thenCA(H)= 0,A(ωF H)=A.

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G, bothCA(H)andA(ωF H)areF G-submodules.Lemma 7yields thatCA(H)= 0

andA=A(ωF H).

Corollary17. LetF be a ﬁeld,charF=p,Aa vector space overF,Ga minimal

artinian subgroup ofGL(F , A). IfHis a nonidentity ﬁnite normalp-subgroup ofGand

Bis a nonzeroF G-submodule ofA, thenCH(B)= 1.

Proof. In fact, if H1 = C_H(B) ≠ 1, then H1 is a nonidentity ﬁnite normal p-subgroup ofG. SinceB≤CA(H1), we obtain a contradiction withCorollary 12.

Corollary₁₈. _Let_F _{be a ﬁeld,}_char_F=_p_,_A_{a vector space over}_F_,_G_{a minimal}

artinian subgroup ofGL(F , A). Furthermore, letHbe a nonidentity normalp-subgroup

having an ascending series ofG-invariant subgroups

1 =H0≤H1≤ ··· ≤Hα≤Hα+1≤ ··· ≤Hγ=H (5)

with ﬁnite factors. IfBis a nonzero properF G-submodule ofA, thenCH(B)= 1.

Proof. _{We use induction on}_{α. If}_α=_{1, then the assertion follows from}_Corollary 13. Letα >1, and we have already proved thatCHβ(B)= 1for allβ < α.

LetCα =CHα(B). If αis a limit ordinal, thenHα=

β<αHβ, and thereforeCα=

β<α(Cα∩Hβ). ButCα∩Hβ=CHβ(B)= 1. ThusCα= 1.

Suppose now thatαis not a limit, and putL=Hα−1. Assume thatCα≠1. Then

Cα∩L=CL(B)= 1, so thatCαCα/(Cα∩L)CαL/L≤Hα/L. It follows thatCαis a

ﬁnite normal subgroup ofG. And we obtain a contradiction withCorollary 12because B≤CA(Cα). HenceCHα(B)= 1. Forα=γwe obtain thatCH(B)= 1.

LetGbe a group. A normal subgroupHis called the hyperﬁnite radical ofGifH satisﬁes the following conditions:

(1) Hpossesses an ascending series ofG-invariant subgroups

1 =H0≤H1≤ ··· ≤Hα≤Hα+1≤ ··· ≤Hγ=H, (6)

every factor of which is ﬁnite;

(2) G/Hhas no nonidentity ﬁnite normal subgroups. We will denote the hyperﬁnite radical ofGbyHF (G).

Let Soc(G)=Xλ∈ΛSλ, whereSαis a minimal normal subgroup ofG,λ∈Λ. Put

Λab=

λ∈Λ|Sλis abelian

, Socab(G)=Xλ∈ΛabSλ. (7)

Corollary₁₉. _Let_F _{be a ﬁeld,}_char_F=_p_,_A_{a vector space over}_{F , G}_{a minimal}

artinian subgroup of GL(F , A). LetS =Socab(G)∩HF (G). Then S is ap-subgroup

including a subgroupQsuch thatS/Qis a locally cyclic group andCoreG(Q)= 1.

Proof. _Clearly_S_{is a subgroup of the locally soluble radical of}_{F C(G). By}_Corollary 17 of Lemma 11, S is a p-subgroup. LetB be a minimal F G-submodule of A. By

Corollary 18,CS(B)= 1. In other words,S is imbedded in an irreducible subgroup

of GL(F , B). And now we can use [1, Lemma 8.2].

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Theorem₂₀. _Let_F_{be a ﬁeld,}_A_{a vector space over}_F_,_G_{a minimal artinian subgroup}

ofGL(F , A). IfGis anF C-group, thenSocab(G)is ap-subgroup including a subgroup

Qsuch thatSocab(G)/Qis a locally cyclic group andCoreG(Q)= 1, wherep=charF.

Proof. _Let_T _{be the periodic part of}_G,_S_{the locally soluble radical of}_{G. For every} elementx∈T, the subgroupxG_{is ﬁnite by Ditsmann’s lemma (cf. [}₇_{, Corollary 2 to}

Lemma 2.14]). This implies the inclusionT≤HF (G). In particular, Socab(G)≤HF (G).

Now we can useCorollary 19ofLemma 15.

The results of [3] imply that for the groupG having the structure, described in

Theorem 20, there is a simpleF G-moduleAsuch thatCG(A)= 1. This means that

this theorem cannot be strengthened. Thus, minimal artinian linearF C-groups have the same structure as irreducible linearF C-groups.

Theorem₂₁. _Let_F_{be a ﬁeld,}_A_{a vector space over}_F_,_G_{a minimal artinian subgroup}

ofGL(F , A). IfGis a hypercentral, thent(ζ(G))is a locally cyclicp-subgroup, where p=charF.

Proof. _By _{Corollary 12} _of _{Lemma 11}_{, the periodic part} _T _{of the group} _G _{is a} p-subgroup. Since G is a hypercentral group, T =HF (G). Choose a minimal F G-submoduleBofA. ByCorollary 18ofLemma 15,T∩CG(B)= 1, that is,TT CG(B)/

CG(B). In other words,Tis imbedded in an irreducible subgroup of GL(F , B). Now we

can useLemma 1.

Corollary ₂₂. _Let_F _{be a ﬁeld,}_A _{a vector space over} _F_, _G _{a minimal artinian}

subgroup ofGL(F , A). IfGis abelian, thent(G)is a locally cyclicp-subgroup, where

p=charF.

Lemmas3,4,5, and6show that, for the groupGhaving the structure described in

Theorem 21(and in its corollary), there is a simpleF G-moduleAsuch thatCG(A)= 1. This means that this theorem (and its corollary) cannot be strengthened. Thus, minimal artinian linear hypercentral (and abelian) groups have the same structure as irreducible linear hypercentral (abelian) groups.

In connection withLemma 2andTheorem 21, there arises the following question: letFbe a locally finite field,Ga hypercentral group of finite 0-rank. LetGbe a minimal artinian subgroup of GL(F , A). Can we claimζ(G)to be periodic? The following simple example gives a negative answer to it.

LetF be a ﬁeld,Aa vector space overF of countable dimension,{an|n∈N}a

basis ofA,xan inﬁnite cyclic group. Deﬁne the action ofxonAby the rule

a1x=a1, an+1x=an+1+an, or a1(x−1)=0, an+1(x−1)=an, n∈N. (8)

Then we can considerAas anFx-module. It is easy to see thatA=A(x−1)and every properFx-submodule ofAcoincides with somea1F+ ··· +anF, n∈N. In

particular, theFx-moduleAis minimal artinian andCx(A)= 1.

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Leonid A. Kurdachenko: Mathematics Department, Dnepropetrovsk University, Provulok Naukovyi13,49050Dnepropetrovsk, Ukraine

E-mail address:[email protected]

Igor Ya. Subbotin: Mathematics Department, National University,9920S. La Cienega Blvd, Inglewood, CA90301, USA