Abstract
Abstract The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ( V ) \mathrm{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 g-\mathrm{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.
Highlights
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 contains SL.n; K/
If K D Fp (p a prime), the scalar matrix diag.x; x; : : : ; x/ for nonzero x belongs to SL.p 1; Fp/
Summary
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 Dsc.n; K/) contains SL.n; K/. Theorem B of [9] says that all proper normal subgroups of GLcf.N; K/ are contained in Dsc.N; K/ GLfr.N; K/. It means that Dsc.N; K/ GLfr.N; K/ is a maximal normal subgroup of GLcf.N; K/ and the corresponding factor group is simple.
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