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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 boundedp-subgroup whenever charF=p, orT is a nilpotent divisible torsion-free subgroup whenever charF =0. Thus, the case of irreducible linear groups is basic.
Irreducible linear groups as the automorphism groups of abelian chief factors, play a crucial role in Group Theory, and their investigation is very useful for the solution of many group theoretical problems. For the infinite-dimensional case, the irreducible groups under some natural restrictions have been studied by Hartley and McDougall [2], Za˘ıcev [13], Robinson and Zhang [9], Franciosi, de Giovanni, and Kurdachenko [1], and Kurdachenko and Subbotin [5].
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 artinian 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=AnnF 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.
This statement is an immediate corollary of the known theorem of I. Schur.
A groupGis said to have finite 0-rankr0(G)=r(or finite torsion-free rank) ifGhas a finite subnormal series with exactlyrinfinite cyclic factors being the others periodic.
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. LetGbe a hypercentral group of finite0-rank,F a locally finite field,Aa 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. LetFbe a field,p=charF,Gan abelian group of infinite0-rank. Ift(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 =charF, 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. LetF be a field,p=charF, Ga 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. LetRbe a ring,Aa minimal artinianR-module. ThenAdoes 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. LetRbe a ring,Aa minimal artinianR-module. ThenSocR(A)is a nonzero proper submodule ofA.
Lemma9. LetF be a field,Ga 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. LetF be a field,Ga group,Aa minimal artinianF 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−1CH(B)x= 1, in particular,H≤Xx∈XH/(x−1CH(B)x).
Proof. By Wilson’s theorem [11]Ais an artinianF H-module. SinceAhas 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−1CH(B)x, then it follows that
x∈Xx−1CH(B)x≤CH(A)= 1. By Remak’s theorem, H≤Xx∈XH/(x−1CH(B)x).
Lemma11. LetF be a field,Ga group,Aa minimal artinianF 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. LetF 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.
Proof. Denote byPthe Sylowp-subgroup ofG, and suppose thatP≠1. Since G is a hypercentral group,P∩ζ(G)≠1. Let 1≠z∈ζ(G)∩P. Since the additive group ofAis an elementary abelianp-group, a natural semidirect productB zis a nilpotent group (cf. [8, Lemma 6.34]). Therefore[Az, Az]=A(z−1)≠A, which contradictsLemma 11. This contradiction shows thatP= 1.
LetGbe a group. Put F C(G)=
x∈G|xG=
g−1xg|g∈G
is 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∈Op(F C(G)). PutY= 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−1Zx≤x−1CG(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.
Proof. ByLemma 15,A=
λ∈ΛMλ,whereMλis a simpleF H-submodule,λ∈Λ.
SinceMλ(ωF H)is anF H-submodule ofMλ, then eitherMλ(ωF H)=MλorMλ(ωF H)
= 0. It implies the equalityA=CA(H)⊕A(ωF H). SinceHis a normal subgroup of
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 = CH(B) ≠ 1, then H1 is a nonidentity finite normal p-subgroup ofG. SinceB≤CA(H1), we obtain a contradiction withCorollary 12.
Corollary18. LetF be a field,charF=p,Aa vector space overF,Ga 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 onα. Ifα=1, then the assertion follows fromCorollary 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. LetF be a field,charF=p,Aa vector space overF , Ga 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. ClearlySis a subgroup of the locally soluble radical ofF C(G). ByCorollary 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].
Now we can expose the main results.
Theorem20. LetFbe a field,Aa vector space overF,Ga 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. LetT be the periodic part ofG,Sthe locally soluble radical ofG. For every elementx∈T, the subgroupxGis 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. LetFbe a field,Aa vector space overF,Ga 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. LetF 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.
Also it shows that the question about the internal structure of minimal artinian modules requires separate consideration.
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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
E-mail address:[email protected]
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