Nilpotency in Groups with the Minimal Condition on Centralizers,

The Sylow-2-subgroups of a periodic group with minimal condition on centralizers are locally finite and conjugate. The same holds for the Sylow-p-subgroups for any prime p, provided the subgroups generated by any two p-elements of the group are finite. In the non-periodic context, the bounded left Engel elements of a group with minimal condition on centralizers form the Fitting subgroup.