  We consider a finiteness condition on centralizers in a group G, namely that
|C_G (x) : <x>| is finite for every non-normal cyclic subgroup <x> of G. For
periodic groups, this is the same as |C_G (x)| is finite for every non-normal
cyclic subgroup <x> of G. We give a full description of locally finite groups
satisfying this condition. As it turns out, they are a special type of cyclic
extensions of Dedekind groups. We also study a variation of our condition,
where the requirement of finiteness is replaced with a bound: |C_G (x) : <x>| <
n for every non-normal cyclic subgroup <x> of G, for some fixed n. In this
case, we are able to extend our analysis to the class of periodic locally
graded groups.
