The structure of groups of finitary transformations
type: Item
instance of: scholarly article
title: The structure of groups of finitary transformations
zbMATH Open document ID: 0669.20031
DOI: 10.1016/0021-8693(88)90068-3
author: Richard E. Phillips
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let V be a vector space over a field k. A linear map g: \(V\to V\) is called finitary, if dim V(g-1)\(<\infty\), or equivalently, if g acts as the identity on a subspace of finite codimension in V. The set of all invertible finitary linear transformations of G is a subgroup FGL(V) of the full group of invertible linear transformations of V. A group is called a k-finitary linear group, if it can be embedded in FGL(V) for some vector space V over k. Examples include (i) any group that can be faithfully represented by matrices of finite size over k (these are here called finite-dimensional k-linear), and (ii) any group of finitary permutations, acting on the associated permutation module over k. A recent and striking theorem of \textit{J. I. Hall} [J. Algebra 119, 337-359 (1988; Zbl 0665.20017)] asserts that if k has characteristic zero and G is an infinite simple locally finite k-finitary linear group, then \(G\cong Alt(\Omega)\) for some set \(\Omega\). The present paper seeks to lay the groundwork for the general study of finitary linear groups, and obtains a lot of information about the locally finite ones, particularly the ``non-modular case'' when the groups concerned have no non-trivial elements of order equal to the characteristic of the underlying field. More general finitary linear groups also receive attention. We now describe the main results in this stimulating paper. Theorem A: Let G be a periodic \((=locally\) finite) k-finitary linear group, and suppose we have the non-modular case (see above). Then (i) G is a subdirect product of irreducible k-finitary linear groups, each of which is an image of G (these are essentially the irreducible constituents of G in a natural sense; note that ``direct product'' means ``restricted direct product'' throughout) (ii) if G is irreducible, then G has a normal subgroup N such that N is a subdirect power of a finite- dimensional k-linear group and G/N is isomorphic to a transitive group of finitary permutations, (iii) i…
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20B07
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20B35
zbMATH DE Number: 4095765
zbMATH Keywords: vector space
zbMATH Keywords: invertible finitary linear transformations
zbMATH Keywords: full group of invertible linear transformations
zbMATH Keywords: group of finitary permutations
zbMATH Keywords: permutation module
zbMATH Keywords: infinite simple locally finite k-finitary linear group
zbMATH Keywords: finitary linear groups
zbMATH Keywords: subdirect product of irreducible k- finitary linear groups
zbMATH Keywords: transitive group of finitary permutations
zbMATH Keywords: minimal subnormal irreducible subgroup
zbMATH Keywords: homogeneous components
zbMATH Keywords: action
zbMATH Keywords: permutation action
reviewed by: Brian Hartley
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90068-3
OpenAlex ID: W2001975835
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description: scientific article; zbMATH DE number 4095765