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Finitary linear groups: A survey. (English) Zbl 0840.20048

Hartley, B. (ed.) et al., Finite and locally finite groups. Proceedings of the NATO Advanced Study Institute held in Istanbul, Turkey, 14-27 August 1994. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 471, 111-146 (1995).
Let \(V\) be a vector space, not necessarily finite-dimensional, over a field \(F\). A linear automorphism \(g\) of \(V\), that is an element of \(GL(V)\), is said to be finitary if \(V(g-1)\) has finite dimension over \(F\). A subgroup \(G\) of \(GL(V)\) is finitary if each of its elements is finitary. Various examples of finitary linear groups have appeared intermittently over the years and finitary permutation groups were in vogue in the 1970’s, but the theory of finitary linear groups really only took off about 10 years ago. A very substantial amount of work has now been done, by a number of people in three continents. We are approaching the point where a book on the subject is needed, to enable those not yet directly involved to learn the subject and to make it easier for those who are involved to quote results. As an interim measure the author in this paper has done valuable service by presenting a survey of the subject up to about 1994. Proofs generally are not included.
The paper starts with some examples of finitary linear groups and the historical background. He then looks at finitary permutation groups (Section 4). These play an important role in the theory. In Section 5 he introduces some basic ideas, particularly the degree function \(\deg(g)=\dim_FV(g-1)\). Section 6 gives some general structure theorems and discusses the notions of irreducibility and unipotence. Section 7 is on imprimitive groups. The author then considers specific group theoretic conditions. Section 8 is on locally soluble groups, Section 10 on locally finite groups (the most developed part of the theory and the source of much work on finitary linear groups) and Section 11 on residual properties. Section 9 considers groups with generating sets of elements of bounded degree. This may seem technical, but it is important for much of the theory. Sections 12 and 13 consider various local problems (beware of the example at the foot of p. 140). The final Section 14 discusses some problems for the future. Many of the same questions can be asked where \(F\) is just a division ring, that it is not necessarily commutative. A great deal has been done on this generalization. Answers exist, but not surprisingly, are more complex. These are only mentioned in passing. The author has had more than enough material to survey just on the finitary linear case. There is a comprehensive bibliography of 83 items.
For the entire collection see [Zbl 0827.00038].

MSC:

20H20 Other matrix groups over fields
20-02 Research exposition (monographs, survey articles) pertaining to group theory
20G15 Linear algebraic groups over arbitrary fields
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