The seriality problem for weakly Sylow-permutable subgroups in locally finite groups (Q2214126)

This paper addresses the following problem. Suppose $H$ is a subgroup of the locally finite group $G$ such that $HP = PH$ for every maximal $p$-subgroup $P$ of $G$ and for every prime $p$ such that $H$ contains an element of order $p$. The problem is the following. Does this imply that $H$ is always serial in $G$? The starting point of the investigation is the 1962 theorem of O. H. Kegel, that if $G$ is also finite, then $H$ is subnormal in $G$. Examples show that in the locally finite case $H$ need not be subnormal or even ascendant, though it is in some very special cases. The problem in general remains unresolved, but here the author produces positive results for some six special cases. These cases tend to involve some sort of finiteness condition (e.g. $H$ is a $p$-group and $G$ satisfies min-$p$) or some sort of weak commutativity (i.e. $H$ hyperfinite and $G$ hyper locally nilpotent). The author also produces short proofs of some previously known cases.