General Linear Transformations

AbstractLet k be an infinite cardinal, F a field, and let GL(k, F) be the group of all non-singular linear transformations on a ki-dimensional vector space V over F. Various examples are given of maximal subgroups of GL(k, F). These include (i) stabilizers of families of subspaces of V which are like filters or ideals on a set, (ii) almost stabilizers of certain subspaces of V, (iii) almost stabilizers of a direct decomposition of V into two k-dimensional subspaces.It is also noted that GL(k, F) is not the union of any chain of length k of proper subgroups.

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General linear transformations of locally convex spaces

Mathematische Annalen ◽

10.1007/bf01362768 ◽

1965 ◽

Vol 159 (5) ◽

pp. 309-328 ◽

Cited By ~ 14

Author(s):

Gottfried K�the

Keyword(s):

General Linear ◽

Locally Convex Spaces ◽

Linear Transformations ◽

Locally Convex ◽

Convex Spaces

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Determination of the Parameters of General Linear Transformations by Precision Experiments

Einstein's Relativity and Beyond - Advanced Series on Theoretical Physical Science ◽

10.1142/9789812813480_0021 ◽

2000 ◽

pp. 273-281

Keyword(s):

General Linear ◽

Linear Transformations

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Exact numerical computation of the rational general linear transformations

10.1117/12.446427 ◽

2002 ◽

Cited By ~ 1

Author(s):

James A. D. W. Anderson

Keyword(s):

Numerical Computation ◽

General Linear ◽

Linear Transformations

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On Primitive Solvable Linear Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-075-6 ◽

1971 ◽

Vol 23 (4) ◽

pp. 679-685 ◽

Cited By ~ 2

Author(s):

C. R. B. Wright

Keyword(s):

Vector Space ◽

Linear Group ◽

General Linear Group ◽

Proper Subgroup ◽

General Linear ◽

Linear Groups ◽

Extension Field ◽

Linear Transformations ◽

Direct Sum

Let V be a vector space over the field K. A group G of K-linear transformations of V onto itself is primitive in case no proper nontrivial subspace of V is G-invariant and V cannot be written as a direct sum of proper subspaces permuted among themselves by G. Equivalently, G is primitive on V in case G is irreducible and is not induced from a proper subgroup.Suprunenko showed [3, Theorem 12, p. 28] that the n-dimensional general linear group GL(n, K) has a solvable primitive subgroup only if(1) there is a divisor, m, of n such that K has an extension field of degree m containing a primitive p-th root of 1 for each prime p dividing n/m.The main result of this note is the converse fact.

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