Normal subgroups of the group of column-finite infinite matrices

The classical result due to Jordan, Burnside, Dickson, says that every normal subgroup of GL(n;K) (K - a field, n>= 3) which is not contained in the center, contains SL(n;K). A. Rosenberg gave description of normal subgroups of GL(V), where V is a vector space of any infinite cardinality dimension. However, in countable case his result is incomplete. We fill this gap giving description of the lattice of normal subgroups of the group of infinite column-finite matrices indexed by positive integers over any field.


Introduction
Description of normal subgroups is a fundamental problem in group theory. The classical result due to Jordan, Burnside, Dickson, says that every normal subgroup of GL(n, K) (K -a field, n ≥ 3) which is not contained in the center, contains SL(n, K). We extend this result, giving description of normal subgroups in the group GL cf (N, K) of invertible column-finite infinite matrices over K indexed by N. Let GL(n, N, K) denote the subgroup of GL cf (N, K) consisting of all matrices which differ from identity matrix E only in first n rows and by SL(n, N, K) the subgroup of GL(n, N, K) containing matrices which in left upper n × n corner have submatrix with determinant 1. We have natural embeddings: GL(n, N, K) ⊆ GL(n + 1, N, K) and SL(n, N, K) ⊆ SL(n + 1, N, K) and so we obtain two new subgroups of GL cf (N, K): By D sc (N, K) we denote the subgroup of all scalar matrices in GL cf (N, K), by H, H 1 we denote the subgroup generated by H and H 1 and by H × H 1 their internal direct product.
We prove the following results: are normal subgroups of GL cf (N, K).
The group SL f r (N, K) and the factor group GL cf (N, K)/(D sc (N, K) × GL f r (N, K)) are simple. The group D sc (N, K) and the factor groups This shows that both Theorems give the complete description of normal subgroups of GL cf (N, K).
The lattice of normal subgroups of GL cf (N, K) "modulo the center" is shown in the figure below (we abbreviate notation for convenience, see the beginning of next section.) The thin line between 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 where V is a vector space of any infinite cardinality dimension ℵ δ . For any infinite cardinal ℵ ν ≤ ℵ δ , by L ν (V ) we denote the set of all linear transformations with ranges of dimension < ℵ nu and let GL ν (V ) be a subgroup of GL(V ) consisting of all elements of the form id + A, A ∈ L ν (V ). The main result (Theorem B) of [4] shows that if N is a normal subgroup of GL(V ), then either N = H · id × GL ν (V ), where 0 < ν ≤ δ and H is a subgroup of K * , or N ≤ K * · id × GL 0 (V ). It is clear that in countable case this result is incomplete. In our terms, he proved only that D sc (N, K) × GL f r (N, K) is maximal normal subgroup of GL cf (N, K) and thus GL cf (N, K)/(D sc (N, K) × GL f r (N, K)) is simple.
Our results fill this gap 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 note that similar description of ideals of the Lie algebra of infinite column-finite matrices over any field was obtained in [3].

Proofs of main results
Let K be any field. We start with the following Lemma Lemma 2.1. D sc (N, K) is the center of GL cf (N, K).
The proof is similar to that for finite dimensional matrices and we will omit it.
Lemma 2.2. The group GL cf (N, K) is generated by the row and columnfinite matrices and upper triangular matrices.
This lemma follows easily from considerations in chapter 2 of [4]. Another proof one can find in [6].
The following Lemma is Theorem 3.3 of [5]. Its proof is contained in chapter 5 of [5]. P. Vermes in [6] gave another proof for K = C, however it can be easily adopted to arbitrary field K (see also [2]). 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 1.1. Now we prove normality of groups listed in Theorem 1. The normality of D sc (N, K) follows from Lemma 1. Let g ∈ GL f r (N, K) and s be a string with shape (n 1 , n 2 , . . .). Then g has a form ĝ ⋆ 0 e whereĝ ∈ GL(n, K) for some n ∈ N and e is the infinite identity matrix. We choose minimal t such that m = t i=1 n i ≥ n. We can extend block decomposition of g to g ⋆ 0 e whereg ∈ GL(m, K) . Then s −1 gs has a form ⋆ ⋆ 0 e and belongs to GL ( m, N, K). If g ∈ SL f r (N, K), then similar arguments show that s −1 gs ∈ SL f r (N, K). Now, 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 gu ∈ GL f r (N, K). The normality of GL f r (N, K) (and SL f r (N, K)) follows from Lemma 2.2 and 2.3. The normality of other subgroups from Theorem 1.1 follows from remark after Theorem 1.2. Proof of Theorem 1.2.
Simplicity of SL f r (N, K) was proved by Clowes and Hirsch in [1]. The fact that the factor group GL cf (N, K)/(D sc (N, K) × GL f r (N, K)) is simple follows from Theorem B of [4].
Let d(α) = (α −1)e 11 + e, where e has 1 in (1, 1) entry and zero elsewhere. For every matrix g ∈ GL f r (N, K) we have a unique decomposition where α = det(ĝ) and d(α −1 ) · g ∈ SL f r (N, K). This shows that GL f r (N, K)/SL f r (N, K) is isomorphic to K * . Now other statements of Theorem 1.2 are obvious.