Sylow subgroups which are maximal in the universal locally finite group of Philip Hall (Q1291107)

P. Hall has shown that there is a countable locally finite group \(U\) such that every finite group embeds into \(U\) and any two isomorphic finite subgroups of \(U\) are conjugate in \(U\). Moreover, \(U\) is a simple group and is unique up to isomorphism [see \textit{P. Hall}, J. Lond. Math. Soc. 34, 305-319 (1959; Zbl 0088.02301) or Chapter 6 of \textit{O. H. Kegel} and the reviewer, Locally finite groups, North-Holland, Amsterdam (1973; Zbl 0259.20001)]. The author of the paper under review shows that for every prime \(p\) the group \(U\) above contains a \(p\)-subgroup that is also a maximal subgroup of \(U\). In particular this also exhibits an infinite locally finite simple group with a locally nilpotent maximal subgroup with no involutions. A number of conditions are known that force a locally finite group with a locally nilpotent maximal subgroup to be (locally) soluble and the author includes a summary of these.