Periodic simple groups of finitary linear transformations. (English) Zbl 1106.20038

This is a long, important and very impressive paper. Those in the know have been waiting for the definitive version for some considerable time, various summaries and drafts having already been in circulation.
The author classifies completely the infinite periodic finitary simple groups. The infinite periodic simple linear groups of finite dimension were classified some 14 years ago, almost simultaneously in four separate papers by five different people. Thus the author’s work focuses on the infinite dimensional finitary groups.
Below \(F\) denotes a (commutative) field and \(V\) an infinite dimensional vector space over \(F\). A finitary group is a subgroup of a finitary general linear group \[ \text{FGL}(V)=\{g\in\text{GL}(V):\dim_FV(g-1)<\infty\}, \] for some \(F\) and \(V\). The main result of this paper is the following.
Theorem. Let \(G\) be a periodic simple subgroup of some \(\text{FGL}(V)\). Then \(G\) is isomorphic to one of the following:
1) A periodic simple linear group of finite dimension.
2) An infinite alternating group.
3) An infinite dimensional finitary symplectic group.
4) An infinite dimensional finitary unitary group.
5) An infinite dimensional finitary orthogonal group.
6) An infinite dimensional finitary analogue of the special linear groups.
As remarked above, groups as in 1) have already been classified. Groups as in 3) to 6) only appear in positive characteristics. Groups 2) to 5) are what one might expect. Those of 6) however are more complex in that there can be many finitary analogues of the finite dimensional special linear groups. They can be elegantly but less briefly described in terms of pairings, when they can also be fitted into the general pattern of the groups 3) to 5). Alternatively we can use transvections. The finite dimensional special linear groups are generated by the transvections and here we can do something similar.
Let \(V\) be an infinite dimensional vector space over the locally finite field \(F\) and let \(W\) be a subspace of \(V^*=\operatorname{Hom}_F(V,F)\) such that \[ \{v\in V:v\varphi=0\text{ for all }\varphi\text{ in }W\}=\{0\}. \] Let \(t_{\varphi x}\colon v\mapsto v+(v\varphi)x\) for all \(v\in V\), where \(x\in V\) and \(\varphi\in V^*\) with \(x\varphi=0\). Then the subgroup of \(\text{GL}(V)\) generated by the transvections \(t_{\varphi x}\) for all \(x\in V\) and \(\varphi\in W\) are the groups of 6). If \(V\) has (infinite) countable dimension for example, then clearly \(W\) can be chosen to have countable or uncountable dimension and this leads to different ‘finitary special linear groups’.
Not surprisingly the proofs are far to long and complex to summarise here.


20H20 Other matrix groups over fields
20F50 Periodic groups; locally finite groups
20E32 Simple groups
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