Block-diagonal characterization of locally finite simple groups of \(p\)-type. (Q2869864)

Let \(G\) be a simple, locally finite group. Then \(G\) always has a Kegel cover, that 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 for every finite subgroup \(X\) of \(G\) there exists at least one pair \((H,M)\) in the Kegel cover such that \(X\) is contained in \(H\) and intersects \(M\) in \(\langle 1\rangle\). This group \(G\) is said to be of \(p\)-type, for \(p\) some prime, if \(G\) is not isomorphic to any finitary linear group and if every Kegel cover of \(G\) contains at least one pair \((H,M)\) with \(H/M\) isomorphic to a classical group defined over a field of characteristic \(p\).NEWLINENEWLINE In earlier work [J. Algebra 315, No. 1, 419-453 (2007; Zbl 1128.20023)] the author proved that if our group \(G\) is of \(p\)-type, then it has at least one Kegel cover of a particularly complex and restricted type.NEWLINENEWLINE In this present paper the author proves the converse of the theorem and uses it to construct a general family of simple, locally finite groups of \(p\)-type.