Locally finite simple groups of 1-type
Basically, this paper is about large locally finite simple groups \(G\), where by large we mean \(G\) has no non-trivial finitary representations over any field, that is, where \(G\) is non-finitary. It is difficult to describe the contents of this paper adequately without giving a large number of technical definitions. We will try to give some idea of its flavour by selecting a few titbits. Let \(G\) be a non-finitary locally finite simple group. Suppose that for every finite subgroup \(F\) of \(G\) there is a finite subgroup \(H\) of \(G\) containing \(F\) and a normal subgroup \(K\) of \(H\) with \(F\cap K=\langle 1\rangle\) such that \(H/K\) is isomorphic to an alternating group. If \(F\) is a finite subgroup of \(G\) for which there is a finite subgroup \(F^*\) of \(G\) containing \(F\) such whenever we have a finite subgroup \(H\) of \(G\) containing \(F^*\) and \(K\) normal in \(H\) with \(H/K\) an alternativing group on \(\Omega\) of degree at least 7, then \(F\) has no regular orbit on \(\Omega\), then the authors describe the normal closure of \(F\) in \(L\) for all large enough finite subgroups \(L\) of \(G\) containing \(F\). They call such an \(F\) non-regular. If \(F\) is any finite subgroup of \(G\), the authors give a necessary and sufficient condition for \(F\) to be non-regular in terms of Kegel covers. Basically it hangs on the existence or otherwise of a bound on the number of regular orbits of \(F\) on any \(\Omega\), where \(H\), \(K\) and \(\Omega\) are as above and \((H,K)\) is in the Kegel cover. Finally, there is a group \(G\) as above, with a non-regular finite subgroup that is not absolutely simple, meaning that \(G\) has a proper serial subgroup.
scientific article; zbMATH DE number 1740524