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c

Journal “Algebra and Discrete Mathematics”

On some linear groups,

having a big family of

G

-invariant subspaces

L. A. Kurdachenko, A. V. Sadovnichenko

A b s t r ac t . Let F be a field, A a vector space over F,

GL(F, A) be the group of all automorphisms of the vector space

A. IfB is a subspace ofA, then denote byBF Gthe G-invariant subspace, generated byB. A subspaceBis called nearlyG-invariant, ifdimF(BF G/B) is finite. In this paper we described the situation

when every subspace ofAis nearlyG-invariant.

Introduction

Let F be a field, A a vector space over F and GL(F, A) a group of all F-automorphisms ofA. If G is a subgroup ofGL(F, A) then, as usual, a subspaceB ofA is calledG-invariant, ifbxB for every bB andxG. The theory of linear groups over finite dimensional space is very well developed. This is one of the most developed group-theoretical theories (see, for example, the books [1, 10, 11]). However, in the case when Ahas infinite dimension over F, the situation became totally different. This case is much more complicated and its consideration requires some additional restrictions. Imposing classical finiteness conditions is one of the most efficient and natural approaches here. The study of infinite dimensional linear groups satisfying some finiteness conditions proved to be very promising. Many valuable results have been obtained in this way (see, for example, the surveys [9, 5]).

Recently began to study another approach in studying of infinite di-mensional linear groups. This approach is based on the notion of invariance

2010 MSC:15A03, 20F16, 20F29.

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of action of a groupG. We have the following simple fact: if every subspace of Ais G-invariant, thenG must be abelian. Consequently the study of infinite dimensional linear groups having very big family of G-invariant subspaces could be fruitful. This has been shown in the papers [2, 3, 7, 8]. In this paper this approach continues to be implemented.

If B is a subspace of A, then BF G = PgGBg is a G-invariant subspace. It follows that BF G is the least G-invariant subspace of A, includingB. The dimension dimF(BF G/B) is called the upper measure

of non G-invariance. If dimF(BF G/B) = 0, then B = BF G. In other

words, every subspace of A is G-invariant. Therefore it is natural to consider a situation, when the upper measures of non G-invariance of all subspaces of Aare finite. A subspace B of a vector spaceA is called nearlyG-invariant if the upper measure of nonG-invariance ofB is finite.

1. Preliminary results

We will need some concepts and results of modules theory. Let R be a ring,A a module overR. Put

T orR(A) ={aA |AnnR(a)6=h0i}.

We observe that if R is an integral domain, then T orR(A) is an R

-submodule. The submodule T orR(A) is called the R-periodic part of A.

An R-module Ais called R-periodic if A=T orR(A). AnR-module A is

calledR-torsion-free, ifT orR(A) =h0i.

It is easy to see that a factor-moduleA/T orR(A) isR-torsion-free.

Let now Gbe a subgroup of GL(F, A), gG. Denote byJ a group ringF X where X=hxiis an infinite cyclic group. Define the action of x on Aby the rule: ax=ag for each elementaA. This action can be continue in a natural way to the action of a ringJ on A, thusA become an J-module. For this case we will say that A is anJ(g) module.

Lemma 1. Let G be a subgroup of GL(F, A) and suppose that every subspace of A is nearly G-invariant. Then A is periodic as J(g)-module for every element gG.

Proof. Ifg has finite order, sayn, then gn1Ann

J(A). Let nowg be

an element of G, having infinite order. Suppose that the result is false; that is, there exists some element aA such that AnnFhgi(a) = {0}. ThenaFhgi ∼=Fhgi, so thataFhgi=LnNaFhgni. Puth=g2andD=

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are infinite. PutC =DF G. By our conditions dimF(C/D) is finite. It

follows thatdimF((CaFhgi)/D) is finite, and hencedimF(aFhgi/(C

aFhgi)) is infinite. On the other hand, CaFhgi = D⊕(CDg). Since C is G-invariant, DgC, so thatDg =CDg. It follows that C=CaFhgi, and we obtain a contradiction. This contradiction proves a result.

Let R be a commutative ring, A an R-module. We define the R -assassinator ofAas the set

AssR(A) ={P |P is a prime ideal of R such that AnnA(P)6=h0i}.

Let nowD be a Dedekind domain,I be an ideal of D. Put

AI ={aA |aIn=h0i f or some nN}.

Clearly,AI is a D-submodule of A. The D-submodule AI is called the

I-component of A. If A = AI, then A is called an I-module. If P is a

prime ideal, then insteadI-submodule we will say sometimes primary submodule.

If A is D-periodic D-module, then A =LPπAP where π = AssD(A)

(see, for example, [6, Corollary 3.8]). Put

I, k(A) ={aA |aIk=h0i}.

It is easy to see that ΩI, k(A) is aD-submodule and

I,1(A)≤ΩI,2(A)≤. . .≤ΩI, k(A)≤. . .

AI =

[

kN

I, k(A).

Further put A[1] = LPπP,1(Ap), where π = AssD(A). If A

is a D-periodic module and B is a non-zero D-submodule of A, then BA[1]6={0}.

Lemma 2. Let G be a subgroup of GL(F, A) and suppose that every subspace ofAis nearlyG-invariant. Letgbe an element ofGand consider

A as J(g)-module. Then A =A[1] +C where C satisfies the following conditions:

(i) C is an F G-submodule of A;

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Proof. Put L = A[1]. We have A = LB for some subspace B. Put C=BF G. SinceB is nearlyG-invariant,dimF(C/B) is finite. We have

C=B⊕(CL). The finiteness ofdimF(C/B) implies thatdimF(CL)

is finite. ClearlyCL=C[1]. The finiteness of dimF(C[1]) implies that

C is artinian J(g)-module (see, for example, [6, Corollary 7.12]).

Let D be a Dedekind domain, A be a simple D-module. Then A∼= D/P for some maximal idealP. We noted that D/Pk andP/Pk+1 are

isomorphic as D-modules for anykN (see, for example, [6, Corollary 1.28]). In particular, theD-module D/Pk is embedded in the D-module D/Pk+1, k N. Therefore we can consider the injective limit of the

family of D-modules{D/Pk |kN}. Put

CP∞=liminj{D/Pk |kN}.

The D-module CP∞ is called the Prüfer P-module. It follows from the

construction thatCP∞is aP-module, moreover,

P, k(CP∞)∼=D D/Pk, kN.

Furthermore,

P, k+1(CP∞)/P, k(CP∞)∼= (D/Pk+1)/(P/Pk+1)∼=D/P.

Hence, if C is a D-submodule of CP∞ and C 6= CP∞, then C =

P, k(CP∞) for some kN. Similarly, if b /∈ ΩP, k−1(CP∞), then

C=bD.

Observe also that a Prüfer P-module is monolithic and its monolith coincides with ΩP, 1(CP∞).

Lemma 3. Let F be a field, J =Fhxi be a group ring of infinite cyclic grouphxi over F andA be an artinian J-module. If dimF(A) is infinite,

thenA includes a F-subspace B, which is not nearly hxi-invariant.

Proof. SincedimF(A) is infinite,Aincludes a Prüfer P-submoduleC for

some maximal idealP of J. Then C generated by elements {cn |nN}

such that cnJ = CnCn+1 = cn+1J, nN and C = SnNCn.

Furthermore, cn+1 ∈/ Cn, nN, so that the elements cn, nN, are

linearly independent. It follows that an F-subspace E, generated by cn, nN, is LnNcnF. Put B = LkNc2kF, then dimF(E/B) is

infinite, therefore dimF(C/B) is infinite. Assume that B is nearly hxi

-invariant. Then B has a finite codimension in D = BFhxi = Bhxi. It follows thatdimF(C/D) is infinite. On the other hand, the equations

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shows that CnD for all nN, so that C = D. This contradiction

proves a result.

Corollary 1. Let G be a subgroup ofGL(F, A) and suppose that every subspace ofAis nearlyG-invariant. Letgbe an element ofGand consider

A as J(g)-module. Then A=A[1] +C where dimF(C) is finite.

Proof. Lemma 2 implies that A = A[1] +C where an F G-submodule C is artinian as anJ(g)-submodule. Lemma 3 shows that in this case dimF(C) must be finite.

In the next Proposition we describe the local structure of vector space, whose subspaces are nearlyG-invariant.

Proposition 1. LetG be a subgroup ofGL(F, A)and suppose that every subspace of A is nearly G-invariant. Letg be an arbitrary element of G. ThenA includes anFhgi-submodule Dsatisfying the following conditions:

(i) dimF(A/D) is finite;

(ii) every subspace of Dis hgi-invariant.

Proof. Put L = A[1]. Using Corollary 1 we obtain that L has finite codimension. For anFhgi-submoduleL we have a direct decomposition L = Lλ∈Λ, where is a simple Fhgi-submodule for every λ ∈ Λ.

Put M = {λ ∈ Λ | dimF() > 1} and B = LλMAλ. Since every

subspace ofA is nearlyG-invariant, it is likewise nearlyhgi-invariant. An application of Lemma 2.1 of paper [7] shows thatdimF(B) is finite. Put

∆ = Λ\M. Since dimF() = 1 for each λ∈ ∆, = aλF for some

elements,λ∈∆. It follows that aλg=αλaλ for some elements

αλF, λ∈∆. Repeating almost word to word the arguments from a

proof of Proposition 2.2 of paper [7], we obtain that there exists a subset Γ ⊆∆ such that αλ = αµ = α for allλ, µ ∈Γ, and a subset ∆\Γ is

finite. It follows that a subspaceD=LλΓ has finite codimension. If

aD, thena=P1jnβλ(j)(j), whereβλ(j)∈F,λ(j)∈Γ, 1≤jn.

We have

ag = X

1≤jn

(βλ(j)(j))g=

X

1≤jn

βλ(j)((j)g) =

X

1≤jn

βλ(j)(αaλ(j)) =

X

1≤jn

αβλ(j)(j)=α

X

1≤jn

βλ(j)(j)=αa.

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Lemma 4. Let G be a subgroup of GL(F, A) and suppose that every subspace of A is nearly G-invariant. Then dimF(aF G) is finite for each

elementaA.

Proof. Let g1 ∈ G and consider A1 = aFhg1i. By Lemma 1 an J(g1

)-module A is periodic, so that AnnJ(a1) 6= {0}. Recall that every

non-zero ideal of J has finite F-dimension. Hence A1∼=Fhxi/AnnJ(a1) has

finite F-dimension. If agA1 for each element gG, then aF G =

A1 and all is proved. Therefore suppose that there exists an element

g2 ∈ G such that ag2 ∈/ A1. Again a subspace aFhg2i has finite F

-dimension, therefore andA2 =A1+aFhg2ialso has finite F-dimension.

By ag2 ∈/ A1 we obtain that ag1F +ag2F = ag1Fag2F. Suppose

that aF G has infinite F-dimension. Then using the above arguments we can find an infinite subset {gn | nN} of elements of G such

that ag1F +. . .+agnF = ag1F. . .agnF for allnN. It follows

that an F-subspace B, generated by agnF, nN, is LnNagnF. Put

C = LkNag2kF, D = LkNag2k−1F, then B = CD and both

dimF(B/C) and dimF(B/D) are infinite. Let E = CF G. Since C is

nearly G-invariant,dimF(E/C) is finite and hencedimF((EB)/C) is

finite. It follows that dimF(B/(EB)) is infinite. On the other hand,

sinceE isG-invariant,a= (ag1)g1−1∈E. Thenag2k−1 ∈E for allkN,

so that DE and henceBE. This contradiction proves a result.

2. Proof of the main results

Now we can prove the following main results of this paper.

Theorem 1. Let G be a subgroup of GL(F, A) and suppose that every subspace of A is nearly G-invariant. Then A includes anF G-submodule

C satisfying the following conditions:

(i) dimF(C) is finite;

(ii) every subspace of A/C is G-invariant.

Proof. If every subspace of A isG-invariant, then all is proved. Suppose now that there are the elementsa1 ∈Aandg1 ∈Gsuch thata1g1∈/ a1F.

Putd1 =a1(g1−1). It readily follows that dimF(a1F +d1F) = 2, that

isa1F+d1F =a1Fd1F. By Lemma 4 anF G-submoduleA1 =a1F G

has finite dimension andd1∈A1. Proposition 1 shows thatAincludes an

Fhg1i-submodule D1 such thatdimF(A/D1) is finite and every subspace

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F-subspace with a property A = D1⊕Y1. Then dimF(Y1) is finite. If

agaF +A1 for each element aD1, then put C =A1+Y1F G. By

Lemma 4dimF(Y1F G) is finite, so that an F G-submoduleC has finite

dimension. Furthermore, byagaF+A1 we can see that every subspace

of A/C is G-invariant. Therefore assume that there exist the elements a2 ∈D1, g2 ∈G such that a2g2 ∈/ a2F+A1. Putd2 =a2(g2−1), then

dimF((a2F+d2F +A1)/A1) = 2. It follows thata2F+d2FA1 ={0}.

In particular, (a2F+d2F)∩(a1F+d1F) ={0}. PutA2 =A1+a2F G. By

Lemma 4dimF(a2F G) is finite, so that and dimF(A2) is finite. By this

choicea2, d2, a1, d1∈A2. Proposition 1 shows thatA includes an Fhg2i

-submoduleD2such thatdimF(A/D2) is finite and every subspace ofD2is

hg2i-invariant. Then the intersectionD1∩D2 has finite codimension. Let

Y2 be a complement toD1∩D2, thendimF(Y2) is finite. IfagaF +A2

for each elementaD1∩D2, then put C =A2+Y2F G. By Lemma 4

dimF(Y2F G) is finite, so that anF G-submoduleC has finite dimension.

Furthermore, byagaF +A2 we can see that every subspace ofA/C is

G-invariant. If not, then there exist the elements a3 ∈D1∩D2, g3 ∈G

such thata3g3 ∈/ a3F +A2. Again put d3 = a3(g3−1) and repeat the

above arguments. Using these arguments, we come to the two possibilities:

(1) this process will finish after finitely many steps, that is we find an F G-submoduleC, having finite dimension, such that every subspace of A/C isG-invariant;

(2) For every positive integer n we find the elements a1, . . . , anA

and g1, . . . , gnGsuch that the following conditions hold:

(a) ajF+djF =ajFdjF, wheredj =aj(gj−1), 1≤jn;

(b) (anFdnF)T(L1≤kn−1(akFdkF)) ={0}, 1≤jn.

Consider the second possibility more detail. PutB =LjNajF,K =

L

jNdjF,E =B+K. Then BK =h0i. It follows that dimF(E/B)

anddimF(E/K) are infinite. SinceB is nearly G-invariant,B has finite

codimension inV =BF G. It follows thatdimF((VE)/B) is finite, and

hencedimF(E/(VE)) is infinite. An equationVE =B⊕(VEK)

shows thatdimF(VEK) is finite. Then there exists a positive integer

tsuch that dt/ VE. On the other hand,dt =at(gt−1). Then from

atBV and the fact thatV is G-invariant we obtain an inclusion

dtV and hence dtVE. This contradiction shows that second

possibility can not appear really, which proves a result.

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Theorem 2. Let G be a subgroup of GL(F, A) and suppose that every subspace of A is nearly G-invariant. Then the following assertions hold:

(i) if char(F) = 0, then G includes a normal abelian torsion-free subgroup Z such that G/Z is isomorphic to subgroup of L×V, where V is a subgroup of multiplicative group of F and L is a subgroup ofGLn(F) for some positive integer n.

(ii) if char(F) =p is a prime, then G includes a normal abelian ele-mentaryp-subgroupZ such that G/Z is isomorphic to subgroup of

L×V, where V is a subgroup of multiplicative group of F and L is a subgroup ofGLn(F) for some positive integer n.

Proof. Theorem 1 shows that A includes an F G-submodule C, having finite dimension, such that every subspace of A/C is G-invariant. Put K =CG(A/C). By Lemma 3.4 of paper [7] V =G/K is isomorphic to

a subgroup of a multiplicative group of a field F. Put now T =CG(C).

Since dimF(C) = n is finite, L = G/T is isomorphic to a subgroup of

finite dimensional linear group GLn(F). Finally, let Z=TK, thenZ

stabilizes the series of

{0} ≤CA.

By a classical result due to Kaluznin (see, for example, [4, Theorem 1.C.1 and Proposition 1.C.3]) Z is either an elementary abelian p-subgroup if char(F) = p > 0, or a torsion-free abelian subgroup ifchar(F) = 0. Finally, by the Remak’s Theorem, we obtain a new embedding ofG/Z in the direct productG/K×G/T =V ×L and the result is proved.

References

[1] Dixon J.D.The structure of linear groups. VAN NOSTRAND: London, 1971. [2] Dixon M.R., Kurdachenko L.A., Otal J.Linear groups with bounded action. Algebra

Colloquium, 2011, Vol.18, N.3, pp. 487-498.

[3] Dixon M.R., Kurdachenko L.A., Otal J. Linear groups with finite dimensional orbits. Ischia Group Theory 2010, Proceedings of the conference in Group Theory, World Scientific, 2012, pp. 131-145.

[4] Kegel O.H., Wehrfritz B.A.F.Locally finite groups. NORTH-HOLLAND: Amster-dam, 1973.

[5] Kurdachenko L.A.On some infinite dimensional linear groups. Note di Matematica, 2010, Vol.30, N.1, pp. 21-36.

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[7] Kurdachenko L.A, Sadovnichenko A.V., Subbotin I.Ya.On some infinite dimen-sional linear groups. Central European Journal of Mathematics, 2009, Vol.7, N. 2, pp. 178-185.

[8] Kurdachenko L.A, Sadovnichenko A.V, Subbotin I. Ya.Infinite dimensional linear groups with a large family of G-invariant subspaces. Commentationes Mathematicae Universitatis Carolinae, 2010, Vol.51, N.4, pp. 551-558.

[9] Phillips R. Finitary linear groups: a survey. ”Finite and locally finite groups”. NATO ASI ser. C 471, Dordrecht, Kluver, 1995, pp. 111-146.

[10] Suprunenko D.A.Matrix groups. NAUKA: Moscow, 1972.

[11] Wehrfritz B.A.F.Infinite linear groups. SPRINGER: Berlin, 1973.

C o n tac t i n f o r m at i o n

L. A. Kurdachenko Department of Algebra and Geometry, School of Mathematics and Mechanics, National Uni-versity of Dnepropetrovsk, Gagarin prospect 72, Dnepropetrovsk 10, 49010

E-Mail: lkurdachenko@i.ua

A. V. Sadovnichenko E-Mail: sadovnichenko.lit@rambler.ru

References

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