Normal subgroups in the group of column-finite infinite matrices

21 Aug 2018  ·  Waldemar Hołubowski, Martyna Maciaszczyk, Sebastian Żurek ·

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. read more

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Group Theory 20E07, 20H20