  The article deals with profinite groups in which centralizers are virtually
procyclic. Suppose that G is a profinite group such that the centralizer of
every nontrivial element is virtually torsion-free while the centralizer of
every element of infinite order is virtually procyclic. We show that G is
either virtually pro-p for some prime p or virtually torsion-free procyclic.
The same conclusion holds for profinite groups in which the centralizer of
every nontrivial element is virtually procyclic; moreover, if G is not pro-p,
then G has finite rank.
