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BY 4.0 license Open Access Published by De Gruyter September 18, 2021

Normal subgroups in the group of column-finite infinite matrices

Waldemar Hołubowski, Martyna Maciaszczyk and Sebastian Zurek
From the journal Journal of Group Theory

Abstract

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ( n , K ) , where 𝐾 is a field and n 3 , which is not contained in the center contains SL ( n , K ) . Rosenberg described the normal subgroups of GL ( V ) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

1 Introduction

The description of normal subgroups is a fundamental problem in group theory. The classical result due to Jordan, Burnside, Dickson, states that every normal subgroup of GL ( n , K ) , where 𝐾 is a field and n 3 , which is not contained in the center (the group of scalar matrices D sc ( n , K ) ) contains SL ( n , K ) . Moreover, the factor group SL ( n , K ) / ( SL ( n , K ) D sc ( n , K ) ) is simple. The latter groups, in the finite field case K = F q , define an infinite family of finite simple groups PSL ( n , F q ) . Of course, we note that SL ( n , K ) D sc ( n , K ) can be nontrivial and depends on 𝑛 and the characteristic of 𝐾. For example, if K = F p (𝑝 a prime), then the scalar matrix diag ( x , x , , x ) for nonzero 𝑥 belongs to SL ( p - 1 , F p ) (we have x p - 1 = 1 ).

In [4], J. Dieudonné extended this result to the case GL ( n , D ) , where n 3 and 𝐷 is any division ring. That paper defines the noncommutative determinant for GL ( n , D ) and investigates the properties of its commutator subgroup. The last paragraph of the paper [4, number 14 on page 45] announces similar results for the infinite-dimensional case, yet contains only three sentences and no proofs.

Let 𝑉 be a linear space of infinite dimension δ over a division ring 𝐷. By End ( V ) , we denote the ring of endomorphisms of 𝑉 and by GL ( V ) its group of units, i.e. the group of all invertible linear transformations on 𝑉.

Dieudonné considers the subgroup C ( V ) of GL ( V ) which is generated by transvections, i.e. elementary transvections and their conjugates, and says that C ( V ) coincides with the commutator subgroup of F ( V ) – the group of linear transformations 𝑔 such that g - id V has finite-dimensional range. Normal subgroups of F ( V ) which are not contained in the center Z ( V ) contain C ( V ) and any proper normal subgroup of C ( V ) is contained in Z ( V ) , i.e. C ( V ) / ( C ( V ) Z ( V ) ) is simple. The proofs of these results announced by Dieudonné are left to the reader.

Here, we will prove the following generalization.

Theorem 1.1

The group C ( V ) is simple. Moreover, if D = F 2 is the two element field, then Z ( V ) is trivial and C ( V ) coincides with F ( V ) .

For any infinite cardinal ν δ , we will denote by F ν the set of all linear transformations with ranges of dimension < ν . It is well known in the literature that all proper nontrivial two-sided ideals of End ( V ) have the form F ν for some 0 ν δ (see [8, Chapter IX]). So we have a chain of two-sided ideals

{ 0 } < F 0 < F 1 < < F δ < End ( V )

corresponding to cardinals 0 < 1 < < δ . It is clear that this result can be generalized to the description of normal subgroups of GL ( V ) , which was done by Rosenberg in [9].

Let Z * be the multiplicative group of the center 𝑍 of the division ring 𝐷. For any infinite cardinal ν δ , by G ν ( 1 ) , we denote the subgroup of GL ( V ) consisting of all elements of the form id V + A , where A F ν by G ν ( 1 ) . Clearly, G ν ( 1 ) are normal subgroups of GL ( V ) . However, in contrast with the description of two-sided ideals in End ( V ) , the group of scalar operators is also a nontrivial normal subgroup in GL ( V ) (in the End ( V ) case, any two-sided ideal containing id V is actually equal to End ( V ) ). Moreover, the group G ν = Z * id V × G ν ( 1 ) is normal in GL ( V ) .

The main result (Theorem B) of [9] shows that if 𝑁 is a normal subgroup of GL ( V ) , then

  1. either N = H id V × G ν ( 1 ) , where 0 < ν δ and H Z * ,

  2. or N Z * id V × G 0 ( 1 ) = G 0 .

This result completely describes the normal subgroups of GL ( V ) which strictly contain Z * id V × G 0 ( 1 ) (case (i)). As noted by Rosenberg, the only normal subgroups that arise in case (i) are due to the ideals of End ( V ) . There is no doubt that the proof of case (i) is correct. For case (ii), Rosenberg recalls Dieudonné’s paper, writing:

“Furthermore, in [5, p. 45] Dieudonné studied the case (ii). He showed there, that either N Z * i d V or N ( G 0 ( 1 ) C ) = C 0 . Moreover, C 0 = G 0 ( 1 ) , the group C 0 / ( C 0 Z * i d V ) is simple, and G 0 ( 1 ) / C 0 is isomorphic to the multiplicative group of 𝐷 made abelian.”

Here Rosenberg defines C 0 = G 0 ( 1 ) C , where 𝐶 is the group generated by elements of class two, i.e. elements of the form id V + A , where A 2 = 0 (see [1, p. 207]), and says (without proof) that C 0 is equal to the commutator subgroup G 0 ( 1 ) of G 0 ( 1 ) . In Dieudonné’s notation,

G 0 ( 1 ) = F ( V ) and G 0 ( 1 ) = ( F ( V ) ) = C ( V ) .

Theorem A of [9] says that GL ( V ) is generated by elements of class 2. It means that C = GL ( V ) and C 0 = G 0 ( 1 ) C = G 0 ( 1 ) . So, unfortunately, the statement C 0 = G 0 ( 1 ) is false and C 0 should be defined in a different way.

The only new result of Rosenberg which extends results of Dieudonné is [9, Lemma 3.7] which says that every normal subgroup of G ν = Z * id V × G ν ( 1 ) is normal in GL ( V ) . So it suffices to find only normal subgroups in Z * id V × G 0 ( 1 ) to complete the description of normal subgroups of GL ( V ) .

If 𝑉 has countable dimension over a field 𝐾 and we fix a basis e 1 , e 2 , , then GL ( V ) corresponds to the group GL cf ( N , K ) of invertible column-finite infinite matrices over 𝐾 indexed by ℕ. Let GL fr ( N , K ) denote the subgroup of GL cf ( N , K ) which consists of all matrices which differ from the identity matrix only in a finite number of rows. By SL fr ( N , K ) , we denote the subgroup of GL fr ( N , K ) consisting of all matrices for which the determinant of the sub-matrix in the left upper corner covering the non-identity rows is equal to 1. By D sc ( N , K ) , we denote the subgroup of all scalar matrices in GL cf ( N , K ) .

It is clear that GL fr ( N , K ) , SL fr ( N , K ) and D sc ( N , K ) correspond to

F ( V ) = G 0 ( 1 ) , C ( V ) = ( G 0 ( 1 ) ) and Z ( V ) = Z * id V

respectively (n.b. both Rosenberg’s and Dieudonné’s definitions are basis independent).

Theorem B of [9] says that all proper normal subgroups of GL cf ( N , K ) are contained in D sc ( N , K ) × GL fr ( N , K ) . It means that D sc ( N , K ) × GL fr ( N , K ) is a maximal normal subgroup of GL cf ( N , K ) and the corresponding factor group is simple. From [9, Lemma 3.7], every normal subgroup of D sc ( N , K ) × GL fr ( N , K ) is normal in GL cf ( N , K ) , some kind of transitivity of normality.

The fact that SL fr ( N , K ) is simple was proved by Clowes and Hirsch in [3] (unknown to Rosenberg). Moreover, this proof shows, as in the finite-dimensional case, that every normal subgroup of GL cf ( N , K ) not contained in D sc ( N , K ) contains SL fr ( N , K ) . In the proof, it was also shown that SL fr ( N , K ) is generated by elementary transvections and their conjugates.

We take the above results and those results on generators of GL cf ( N , K ) given in [6, 11, 12] to prove the following.

Theorem 1.2

The subgroups

  • D sc ( N , K ) ,

  • SL fr ( N , K ) ,

  • GL fr ( N , K ) ,

  • D sc ( N , K ) × SL fr ( N , K ) ,

  • D sc ( N , K ) × GL fr ( N , K )

are normal subgroups of GL cf ( N , K ) .

The group SL fr ( N , K ) and the factor group

GL cf ( N , K ) / ( D sc ( N , K ) × GL fr ( N , K ) )

are simple. The group D sc ( N , K ) and the factor groups

  • ( D sc ( N , K ) × SL fr ( N , K ) ) / SL fr ( N , K ) ,

  • ( D sc ( N , K ) × GL fr ( N , K ) ) / GL fr ( N , K ) ,

  • GL fr ( N , K ) / SL fr ( N , K ) ,

  • ( D sc ( N , K ) × GL fr ( N , K ) ) / ( D sc ( N , K ) × SL fr ( N , K ) )

are isomorphic to K * .

We note the following consequences of this theorem.

Corollary 1.3

The following statements hold.

  1. The group GL cf ( N , K ) is perfect, i.e. it coincides with its commutator subgroup.

  2. D sc ( N , K ) is the center of GL cf ( N , K ) .

  3. SL fr ( N , K ) is the commutator subgroup of GL fr ( N , K ) .

Since D sc ( N , K ) is the center of GL cf ( N , K ) (by Lemma 2.4), every subgroup of D sc ( N , K ) is normal in GL cf ( N , K ) . Moreover, any subgroup 𝐻 such that SL fr ( N , K ) H GL fr ( N , K ) is normal in GL fr ( N , K ) (Lemma 2.5).

The lattice of normal subgroups of GL cf ( N , K ) “modulo the center” is shown in the figure below (we abbreviate notation for convenience). The thin line between the subgroups H 1 and H 2 ( H 1 < H 2 ) means that the factor group H 2 / H 1 is simple; the thick line means that the factor group H 2 / H 1 is isomorphic to K * .

Remark 1

If 𝐾 is a two element field F 2 , then D sc is trivial, GL fr = SL fr and this lattice reduces in obvious way. Moreover, GL fr ( N , F 2 ) is the only proper nontrivial normal subgroup of GL cf ( N , F 2 ) .∎

It is clear that Theorem 1.2 reduces the problem of determining the normal subgroups of GL cf ( N , K ) to the following problem

Problem

Describe all normal subgroups of the group D sc ( N , K ) × GL fr ( N , K ) which contain SL fr ( N , K ) .

The partial answer is given by the following proposition.

Proposition 1.4

If H 1 D sc ( N , K ) and H 2 is such that

SL fr ( N , K ) H 2 GL fr ( N , K ) ,

then H 1 × H 2 is a normal subgroup of D sc ( N , K ) × GL fr ( N , K ) .

Remark 2

Rosenberg omitted many normal subgroups that are described in Proposition 1.4. He only noted [9, beginning of paragraph 3] that H 1 × GL fr ( N , K ) are normal in GL cf ( N , K ) if H 1 D sc ( N , K ) , but he omitted all subgroups of the form H 1 × H , where SL fr ( N , K ) H < GL fr ( N , K ) (with the exception of SL fr ( N , K ) of course).∎

Moreover, we illustrate the situation with an example which shows that there exist solutions of our problem which are not direct products.

Example

From Theorem 1.2, it follows D sc ( N , K ) K * . Let 𝑓 be a natural homomorphism f : GL fr ( N , K ) GL fr ( N , K ) / SL fr ( N , K ) K * . Let

H = { ( a , b ) D sc ( N , K ) × GL fr ( N , K ) : f ( b ) = a } .

One can easily check that 𝐻 is a normal subgroup in D sc ( N , K ) × GL fr ( N , K ) . Since H D sc ( N , K ) is trivial and H SL fr ( N , K ) = SL fr ( N , K ) , we deduce that 𝐻 is not a direct product. In fact, 𝐻 is a subdirect product. ∎

The main idea in solving the problem uses the fact that if 𝐻 is subgroup of abelian group 𝐺 and φ : H φ ( H ) G is an isomorphism, then

Δ φ ( H ) = { ( g , φ ( g ) ) : g H }

is a subgroup of G × G .

By π i , we denote the natural projection onto the 𝑖-th component of a direct product. The solution to our problem is now summarized in the following theorem.

Theorem 1.5

If 𝐻 is a normal subgroup of D sc ( N , K ) × GL fr ( N , K ) containing SL fr ( N , K ) , then we have that 𝐻 is determined in a unique way by a quintuple ( H 1 , F 1 , H 2 , F 2 , φ ) , where H 1 = H D sc ( N , K ) , H 2 = H GL fr ( N , K ) , F i = π i ( H ) , φ : F 1 / H 1 F 2 / H 2 is an isomorphism and ( x , y ) H if and only if φ ( x H 1 ) = y H 2 .

It is clear that the quintuple ( H 1 , H 1 , H 2 , H 2 , φ ) corresponds to H = H 1 × H 2 . In general, 𝐻 is only a subdirect product of π 1 ( H ) × π 2 ( H ) (which is a fiber product by Goursat’s lemma [5]).

In the case where 𝑉 has countable dimension over a field 𝐾, it is clear that Rosenberg omitted many normal subgroups described in Proposition 1.4 which are of direct product form and all normal subgroups which are not of direct product form (see the example and Theorem 1.5). Moreover, Rosenberg and Dieudonné did not note the triviality of the normal subgroup D sc ( N , K ) SL fr ( N , K ) and the simplification of the description in the case K = F 2 (Remark 1, Theorem 1.1). So Rosenberg’s results are incomplete.

Theorems 1.2 and 1.5 fill these gaps giving the full description of the lattice of normal subgroups of the group of infinite column-finite matrices indexed by positive integers over any field.

We believe that, for 𝑉 not of countable dimension, the description of the normal subgroups of GL ( V ) contained in Z * id V × G 0 ( 1 ) is similar, but it requires more technical work (there are no references like [2, 3, 6, 11, 12] for the countable case over a field). Since every finite division ring is a field by Wedderburn’s theorem, our results hold also for finite division rings.

We note that a similar description of the ideals of the Lie algebra of infinite column-finite matrices over any field was obtained in [7].

2 Proofs of main results

Proof of Theorem 1.1

Assume that C ( V ) / ( C ( V ) Z ( V ) ) is simple. It is clear that the center Z ( V ) = Z * id V of GL ( V ) contains only homotheties, and for every non-identity homothety 𝑔, the element g - id V has infinite-dimensional range if 𝑉 is infinite dimensional and D F 2 . If D = F 2 , then the center is trivial. So C ( V ) Z ( V ) is always trivial and C ( V ) is simple for all division rings. ∎

Let 𝐾 be any field. We start with the following lemmas.

Lemma 2.1

If the subgroup A 1 is normal in the group 𝐴 and the subgroup B 1 is normal in the group 𝐵, then A 1 × B 1 is normal in A × B and

( A × B ) / ( A 1 × B 1 ) ( A / A 1 ) × ( B / B 1 ) .

The above lemma is [10, Theorem 2.30].

Lemma 2.2

The group GL cf ( N , K ) is generated by the row and column-finite matrices and the upper triangular matrices.

This lemma follows easily from the considerations in [9, Chapter 2]. Another proof can be found in [12].

An infinite block-diagonal matrix with finite blocks of sizes n 1 × n 1 , n 2 × n 2 , … is called a string with the shape ( n 1 , n 2 , ) . Of course, a string with shape ( 1 , 1 , ) is a diagonal matrix. The following lemma is [11, Theorem 3.3]. Its proof is contained in [11, Chapter 5]. P. Vermes in [12] gave another proof for K = C ; however, it can be easily adapted to an arbitrary field 𝐾 (see also [6]). By GL rcf ( N , K ) , we denote the subgroup of GL cf ( N , K ) consisting of the row and column-finite matrices.

Lemma 2.3

The group GL rcf ( N , K ) is generated by strings.

Proof of Theorem 1.2

Now we prove the normality of the groups listed in Theorem 1.2. We note that, by [9, Lemma 3.7], it suffices to prove normality in

D sc ( N , K ) × GL fr ( N , K ) .

The normality of D sc ( N , K ) follows from Lemma 2.1. Let g SL fr ( N , K ) , and let 𝑠 be a string with shape ( n 1 , n 2 , ) . Then 𝑔 has the form

( g ^  0 e ) ,

where g ^ SL ( n , K ) for some n N and 𝑒 is the infinite identity matrix. We choose the minimal 𝑡 such that m = i = 1 t n i n . We can extend the block decomposition of 𝑔 to

( g ~  0 e ) ,

where g ~ SL ( m , K ) . Then s - 1 g s has the form

( g ^  0 e ) ,

where g ^ SL ( m , K ) . If 𝑢 is any upper triangular matrix, then we can use on 𝑢 the same block structure as on 𝑔. Simple calculations show that u - 1 g u SL fr ( N , K ) . Now, if h = h 1 h 2 h n is the product of upper triangular and block-diagonal matrices, then we deduce h - 1 g h SL fr ( N , K ) by induction on 𝑛. So SL fr ( N , K ) is normal in GL cf ( N , K ) and of course in GL fr ( N , K ) . The normality of other subgroups from Theorem 1.2 follows from Lemma 2.1.

The simplicity of SL fr ( N , K ) was proved by Clowes and Hirsch in [3]. Moreover, this proof shows, as in the finite-dimensional case, that every normal subgroup of GL cf ( N , K ) which is not contained in D sc ( N , K ) contains SL fr ( N , K ) . The fact that the factor group GL cf ( N , K ) / ( D sc ( N , K ) × GL fr ( N , K ) ) is simple follows from [9, Theorem B].

Let d ( α ) = ( α - 1 ) e 11 + e , where e 11 has a one in the ( 1 , 1 ) entry and zeros elsewhere. For every matrix g GL fr ( N , K ) , we have a unique decomposition

g = d ( α ) ( d ( α - 1 ) g ) ,

where α = det ( g ^ ) and d ( α - 1 ) g SL fr ( N , K ) . This shows that

GL fr ( N , K ) / SL fr ( N , K )

is isomorphic to K * . Now the other statements of Theorem 1.2 are obvious. ∎

Lemma 2.4

D sc ( N , K ) is the center of GL cf ( N , K ) .

As for finite-dimensional matrix groups, we first show that nondiagonal entries of the matrix from the center are equal to 0, then we show that diagonal entries are equal. We will omit the details.

Lemma 2.5

A nontrivial subgroup 𝐻 of GL fr ( N , K ) is normal if and only if it contains SL fr ( N , K ) .

Proof of Lemma 2.5

If 𝐻 is a nontrivial normal subgroup of GL fr ( N , K ) , then from [3] it follows that H SL fr ( N , K ) . Since GL fr ( N , K ) / SL fr ( N , K ) K * is abelian,

( GL fr ( N , K ) ) = [ GL fr ( N , K ) , GL fr ( N , K ) ] SL fr ( N , K )

holds. If SL fr ( N , K ) H GL fr ( N , K ) , then

[ GL fr ( N , K ) , H ] [ GL fr ( N , K ) , GL fr ( N , K ) ] SL fr ( N , K ) H .

So 𝐻 is normal in GL fr ( N , K ) . ∎

Proof of Corollary 1.3

(1) In [2], it was shown that the upper triangular matrix a = ( a i j ) such that a i i = a i , i + 2 = 1 for all i N and a i j = 0 otherwise belongs to the commutator subgroup of the group of upper unitriangular matrices UT ( N , K ) . It does not belong to GL fr ( N , K ) . As ( UT ( N , K ) ) ( GL cf ( N , K ) ) , the group GL cf ( N , K ) is perfect.

(2) follows from Lemma 2.4.

(3) Simple calculations show that ( GL fr ( N , K ) ) SL fr ( N , K ) . The reverse inclusion follows from Lemma 2.5. ∎

Proof of Theorem 1.5

Let 𝐻 be a normal subgroup of D sc ( N , K ) × GL fr ( N , K ) containing SL fr ( N , K ) . So π 1 ( H ) is normal in D sc ( N , K ) , π 2 ( H ) is normal in GL fr ( N , K ) and 𝐻 is a subdirect product of π 1 ( H ) × π 2 ( H ) . The subgroups π 2 ( H ) are in one-to-one correspondence with subgroups of K * by canonical homomorphism GL fr ( N , K ) GL fr ( N , K ) / SL fr ( N , K ) .

So our problem is equivalent to the problem of describing all the subgroups 𝐻 of the abelian group A 1 × A 2 , where A 1 A 2 K * . Let H i = H A i . Since H 1 × H 2 H , we can factorize ( A 1 × A 2 ) / ( H 1 × H 2 ) and reduce by Lemma 2.1 the problem to finding subgroups F ( A 1 / H 1 ) × ( A 2 / H 2 ) such that

F i = F ( A i / H i ) is trivial .

Let K i = π i ( F ) .

From the definition of the F i , it follows that, for every x K 1 , there exists a unique y K 2 such that ( x , y ) F . Moreover, for every y K 2 , there exists a unique x K 1 such that ( x , y ) F . So the mapping x y is an isomorphism of K 1 with K 2 .

Now assume that we have H i F i A i and an isomorphism

φ : F 1 / H 1 F 2 / H 2 .

Then there exists a unique subgroup 𝐻 such that H i = H A i , F i = π ( H ) and ( x , y ) H if and only if φ ( x H 1 ) = y H 2 .

Applying the above considerations in our case, we complete the proof. ∎

Acknowledgements

We are very grateful to A. V. Stepanov for a fruitful discussion and to an anonymous referee for valuable remarks.

  1. Communicated by: Christopher W. Parker

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Received: 2019-12-12
Revised: 2021-08-14
Published Online: 2021-09-18
Published in Print: 2022-03-01

© 2021 Hołubowski, Maciaszczyk and Zurek, published by De Gruyter

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