Groups in which every non-cyclic subgroup contains its centralizer

A subgroup of a group is self-centralizing if the centralizer is contained in . Let denote the class of all groups, in which all non-trivial subgroups are self-centralizing, by . We consider some special families of groups in namely nilpotent groups and süper solvable groups are studied. And we focus on the class of groups with the property that if is a non-cyclic subgroup of , then . It is trivial that all cyclic groups are in the class and class is contained in the class . We also study the finite p-groups and finite simple groups in . Morever, we study the examples about these subjects and give some different examples.