Irreducible and transitive locally-nilpotent by Abelian finitary groups (Q1977794)

Suppose \(G\) is either a transitive finitary permutation group on an infinite set or an irreducible group of finitary linear transformations of an infinite-dimensional vector space over some division ring \(D\) (for example, \(D\) any field). If \(G\) is locally-nilpotent by Abelian, the author proves that \(G\) is a locally finite \(p\)-group for some prime \(p\) (not \(\text{char }D\) in the second case). As corollaries to this the author deduces that \(G\) is also a \(p\)-group if any one of the following three holds. (a) \(G\) is locally supersoluble. (b) The Hirsch-Plotkin radical \(H\) of \(G\) is non-trivial and \(G/H\) satisfies a non-trivial law. (c) \(G/H\) is hypercyclic.