The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of , where 𝐾 is a field and , which is not contained in the center contains . Rosenberg described the normal subgroups of , 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 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.
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 , where 𝐾 is a field and , which is not contained in the center (the group of scalar matrices ) contains . Moreover, the factor group is simple. The latter groups, in the finite field case , define an infinite family of finite simple groups . Of course, we note that can be nontrivial and depends on 𝑛 and the characteristic of 𝐾. For example, if (𝑝 a prime), then the scalar matrix for nonzero 𝑥 belongs to (we have ).
In , J. Dieudonné extended this result to the case , where and 𝐷 is any division ring. That paper defines the noncommutative determinant for 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 over a division ring 𝐷. By , we denote the ring of endomorphisms of 𝑉 and by its group of units, i.e. the group of all invertible linear transformations on 𝑉.
Dieudonné considers the subgroup of which is generated by transvections, i.e. elementary transvections and their conjugates, and says that coincides with the commutator subgroup of – the group of linear transformations 𝑔 such that has finite-dimensional range. Normal subgroups of which are not contained in the center contain and any proper normal subgroup of is contained in , i.e. is simple. The proofs of these results announced by Dieudonné are left to the reader.
Here, we will prove the following generalization.
The group is simple. Moreover, if is the two element field, then is trivial and coincides with .
For any infinite cardinal , we will denote by the set of all linear transformations with ranges of dimension . It is well known in the literature that all proper nontrivial two-sided ideals of have the form for some (see [8, Chapter IX]). So we have a chain of two-sided ideals
corresponding to cardinals . It is clear that this result can be generalized to the description of normal subgroups of , which was done by Rosenberg in .
Let be the multiplicative group of the center 𝑍 of the division ring 𝐷. For any infinite cardinal , by , we denote the subgroup of consisting of all elements of the form , where by . Clearly, are normal subgroups of . However, in contrast with the description of two-sided ideals in , the group of scalar operators is also a nontrivial normal subgroup in (in the case, any two-sided ideal containing is actually equal to ). Moreover, the group is normal in .
The main result (Theorem B) of  shows that if 𝑁 is a normal subgroup of , then
either , where and ,
This result completely describes the normal subgroups of which strictly contain (case (i)). As noted by Rosenberg, the only normal subgroups that arise in case (i) are due to the ideals of . 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 or . Moreover, , the group is simple, and is isomorphic to the multiplicative group of 𝐷 made abelian.”
Here Rosenberg defines , where 𝐶 is the group generated by elements of class two, i.e. elements of the form , where (see [1, p. 207]), and says (without proof) that is equal to the commutator subgroup of . In Dieudonné’s notation,
Theorem A of  says that is generated by elements of class 2. It means that and . So, unfortunately, the statement is false and 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 is normal in . So it suffices to find only normal subgroups in to complete the description of normal subgroups of .
If 𝑉 has countable dimension over a field 𝐾 and we fix a basis , then corresponds to the group of invertible column-finite infinite matrices over 𝐾 indexed by ℕ. Let denote the subgroup of which consists of all matrices which differ from the identity matrix only in a finite number of rows. By , we denote the subgroup of 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 , we denote the subgroup of all scalar matrices in .
It is clear that , and correspond to
respectively (n.b. both Rosenberg’s and Dieudonné’s definitions are basis independent).
Theorem B of  says that all proper normal subgroups of are contained in . It means that is a maximal normal subgroup of and the corresponding factor group is simple. From [9, Lemma 3.7], every normal subgroup of is normal in , some kind of transitivity of normality.
The fact that is simple was proved by Clowes and Hirsch in  (unknown to Rosenberg). Moreover, this proof shows, as in the finite-dimensional case, that every normal subgroup of not contained in contains . In the proof, it was also shown that is generated by elementary transvections and their conjugates.
The group and the factor group
are simple. The group and the factor groups
We note the following consequences of this theorem.
The following statements hold.
The group is perfect, i.e. it coincides with its commutator subgroup.
is the center of .
is the commutator subgroup of .
The lattice of normal subgroups of “modulo the center” is shown in the figure below (we abbreviate notation for convenience). The thin line between the subgroups and ( ) means that the factor group is simple; the thick line means that the factor group is isomorphic to .
If 𝐾 is a two element field , then is trivial, and this lattice reduces in obvious way. Moreover, is the only proper nontrivial normal subgroup of .∎
It is clear that Theorem 1.2 reduces the problem of determining the normal subgroups of to the following problem
Describe all normal subgroups of the group which contain .
The partial answer is given by the following proposition.
If and is such that
then is a normal subgroup of .
Rosenberg omitted many normal subgroups that are described in Proposition 1.4. He only noted [9, beginning of paragraph 3] that are normal in if , but he omitted all subgroups of the form , where (with the exception of of course).∎
Moreover, we illustrate the situation with an example which shows that there exist solutions of our problem which are not direct products.
From Theorem 1.2, it follows . Let 𝑓 be a natural homomorphism . Let
One can easily check that 𝐻 is a normal subgroup in . Since is trivial and , 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 is an isomorphism, then
is a subgroup of .
By , 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.
If 𝐻 is a normal subgroup of containing , then we have that 𝐻 is determined in a unique way by a quintuple , where , , , is an isomorphism and if and only if .
It is clear that the quintuple corresponds to . In general, 𝐻 is only a subdirect product of (which is a fiber product by Goursat’s lemma ).
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 and the simplification of the description in the case (Remark 1, Theorem 1.1). So Rosenberg’s results are incomplete.
We believe that, for 𝑉 not of countable dimension, the description of the normal subgroups of contained in 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 .
2 Proofs of main results
Proof of Theorem 1.1
Assume that is simple. It is clear that the center of contains only homotheties, and for every non-identity homothety 𝑔, the element has infinite-dimensional range if 𝑉 is infinite dimensional and . If , then the center is trivial. So is always trivial and is simple for all division rings. ∎
Let 𝐾 be any field. We start with the following lemmas.
If the subgroup is normal in the group 𝐴 and the subgroup is normal in the group 𝐵, then is normal in and
The above lemma is [10, Theorem 2.30].
The group is generated by the row and column-finite matrices and the upper triangular matrices.
An infinite block-diagonal matrix with finite blocks of sizes , , … is called a string with the shape . Of course, a string with shape is a diagonal matrix. The following lemma is [11, Theorem 3.3]. Its proof is contained in [11, Chapter 5]. P. Vermes in  gave another proof for ; however, it can be easily adapted to an arbitrary field 𝐾 (see also ). By , we denote the subgroup of consisting of the row and column-finite matrices.
The group is generated by strings.
Proof of Theorem 1.2
The normality of follows from Lemma 2.1. Let , and let 𝑠 be a string with shape . Then 𝑔 has the form
where for some and 𝑒 is the infinite identity matrix. We choose the minimal 𝑡 such that . We can extend the block decomposition of 𝑔 to
where . Then has the form
where . If 𝑢 is any upper triangular matrix, then we can use on 𝑢 the same block structure as on 𝑔. Simple calculations show that . Now, if is the product of upper triangular and block-diagonal matrices, then we deduce by induction on 𝑛. So is normal in and of course in . The normality of other subgroups from Theorem 1.2 follows from Lemma 2.1.
The simplicity of was proved by Clowes and Hirsch in . Moreover, this proof shows, as in the finite-dimensional case, that every normal subgroup of which is not contained in contains . The fact that the factor group is simple follows from [9, Theorem B].
Let , where has a one in the entry and zeros elsewhere. For every matrix , we have a unique decomposition
where and . This shows that
is isomorphic to . Now the other statements of Theorem 1.2 are obvious. ∎
is the center of .
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.
A nontrivial subgroup 𝐻 of is normal if and only if it contains .
Proof of Lemma 2.5
If 𝐻 is a nontrivial normal subgroup of , then from  it follows that . Since is abelian,
holds. If , then
So 𝐻 is normal in . ∎
Proof of Corollary 1.3
(1) In , it was shown that the upper triangular matrix such that for all and otherwise belongs to the commutator subgroup of the group of upper unitriangular matrices . It does not belong to . As , the group is perfect.
(2) follows from Lemma 2.4.
(3) Simple calculations show that . The reverse inclusion follows from Lemma 2.5. ∎
Proof of Theorem 1.5
Let 𝐻 be a normal subgroup of containing . So is normal in , is normal in and 𝐻 is a subdirect product of . The subgroups are in one-to-one correspondence with subgroups of by canonical homomorphism .
So our problem is equivalent to the problem of describing all the subgroups 𝐻 of the abelian group , where . Let . Since , we can factorize and reduce by Lemma 2.1 the problem to finding subgroups such that
From the definition of the , it follows that, for every , there exists a unique such that . Moreover, for every , there exists a unique such that . So the mapping is an isomorphism of with .
Now assume that we have and an isomorphism
Then there exists a unique subgroup 𝐻 such that , and if and only if .
Applying the above considerations in our case, we complete the proof. ∎
We are very grateful to A. V. Stepanov for a fruitful discussion and to an anonymous referee for valuable remarks.
Communicated by: Christopher W. Parker
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