Some results concerning simple locally finite groups of 1-type.
Let \(G\) be an infinite simple, locally finite group. Then \(G\) has a Kegel cover; that is, a collection of finite subgroups \(H\) with a specified maximal normal subgroup \(M\) such that for every finite subgroup \(F\) of \(G\) there is some pair \((H,M)\) in the Kegel cover such that \(F\leq H\) and \(F\cap M=\langle 1\rangle\). The group is said to be Type 1 if every Kegel cover of \(G\) contains a pair \((H,M)\) with \(H/M\) an alternating group. This is a highly technical paper and I cannot do better to summarize its contents than to quote the authors' abstract: Several aspects of infinite simple locally finite groups of 1-type are considered. In the first part, the classes of diagonal limits of finite alternating groups, of diagonal limits of finite direct products of alternating groups, and of absolutely simple groups of 1-type are distinguished from each other. In the second part, inductive systems of representations over fields of characteristic zero (which are known to correspond to ideals in the group algebra) are studied in general for groups of 1-type. The roles of primitive respectively imprimitive representations in inductive systems are investigated. Moreover it is shown that in any proper inductive system the depths of the representations of certain alternating subgroups are bounded.
scientific article; zbMATH DE number 2173128
scientific article