: In this thesis investigate number of centralizers in finite groups .Let G be a group and Cent(G) denote the set of centralizers of single elements of G. Let n ? 1 be an integer, that is Cent(G) ={C G (g) | g ? G} where C G (g) is the centralizer of the element g in G. A group G is called n-centralizer if |Cent(G)| = n and primitive n-centralizer if , where Z(G) denotes the centre of G. In this thesis for a finite group G, we give some interesting relations between |Cent(G)| and the maximum number of the pairwise non-commuting elements in G.
