J. Aust. Math. Soc.  81 (2006), 35-47
The structure of groups whose subgroups are permutable-by-finite

M. De Falco
  Dipartimento di Matematica e Applicazioni
  Università di Napoli Federico II
  Complesso Universitario Monte S. Angelo
  Via Cintia
  I–80126 Napoli
  Italy
  mdefalco@unina.it
F. de Giovanni
  Dipartimento di Matematica e Applicazioni
  Università di Napoli Federico II
  Complesso Universitario Monte S. Angelo
  Via Cintia
  I–80126 Napoli
  Italy
  degiovan@unina.it
C. Musella
  Dipartimento di Matematica e Applicazioni
  Università di Napoli Federico II
  Complesso Universitario Monte S. Angelo
  Via Cintia
  I–80126 Napoli
  Italy
  cmusella@unina.it
and
Y. P. Sysak
  Institute of Mathematics
  Ukrainian National Academy of Sciences
  vul. Tereshchenkivska 3
  01601 Kiev
  Ukraine
  sysak@imath.kiev.ua


Abstract
A subgroup $H$ of a group $G$ is said to be permutable if $HX=XH$ for each subgroup $X$ of $G$, and the group $G$ is called quasihamiltonian if all its subgroups are permutable. We shall say that $G$ is a $QF$-group if every subgroup $H$ of $G$ contains a subgroup $K$ of finite index which is permutable in $G$. It is proved that every locally finite $QF$-group contains a quasihamiltonian subgroup of finite index. In the proof of this result we use a theorem by Buckley, Lennox, Neumann, Smith and Wiegold concerning the corresponding problem when permutable subgroups are replaced by normal subgroups: if $G$ is a locally finite group such that $H/H_G$ is finite for every subgroup $H$, then $G$ contains an abelian subgroup of finite index.
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