Block-diagonality of LFS-groups of \(p\)-type. (Q2456191)

This paper is a further contribution to the on-going programme to classify all locally finite simple groups. Here the author proves that a locally finite simple group of \(p\)-type has a Kegel cover of a very restrictive type, this being the main result of his Ph.D. thesis. It would take far too long to explain all the terms involved here and would probable convey little to those not already heavily involved with the project, but here is some idea of the main concepts. A Kegel cover of our locally finite simple group \(G\) is a set of pairs \((H,M)\), where \(H\) is a finite subgroup of \(G\) and \(M\) is a maximal normal subgroup of \(H\), such that every finite subset of \(G\) slots into at least one (and hence many) of the factors \(H/M\). Every locally finite simple group has a Kegel cover. Our group \(G\) is of \(p\)-type for some prime \(p\) if \(G\) is not isomorphic to a finitary group and if every Kegel cover of \(G\) contains at least one pair \((H,M)\) with \(H/M\) a (perfect simple) finite classical group in characteristic \(p\).