Normal subgroups in the group of column-finite infinite matrices

TL;DR

This paper characterizes the lattice of normal subgroups in the group of infinite column-finite matrices over any field, extending classical results to an infinite-dimensional setting.

Contribution

It completes the proof for countably dimensional vector spaces, providing a full description of normal subgroups in the group of infinite column-finite matrices.

Findings

Describes the lattice of normal subgroups in the infinite column-finite matrices group

Extends classical finite-dimensional results to infinite-dimensional case

Fills a gap in Rosenberg's earlier work for countably dimensional spaces

Abstract

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of $GL(n, K)$ ($K$ - a field, $n \geq 3$) 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 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 not complete. 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.