On finitary linear groups with a locally nilpotent maximal subgroup (Q5943637)

Let \(G\) be a locally finite group with a locally nilpotent maximal subgroup \(M\) such that if \(M\) modulo the intersection \(N\) of its conjugates in \(G\) is a 2-group, then \(M/N\) is nilpotent of class at most 2. The authors' main result is the following Theorem. Suppose in addition that \(G\) is a finitary linear group containing no non-trivial element of order the ground-field characteristic. Then \(G\) is locally soluble. As a consequence, and also as a step in the proof of the theorem, if \(G\) is a finitary permutation group and has a subgroup \(M\) as above, then again \(G\) is locally soluble. In a note added in proof, the authors state that F. Leinen and O. Puglisi have succeeded in eliminating the non-modular condition (that is, the condition on the characteristic) from the theorem.