For a group G, | Cent(G) | denotes the number of distinct centralizers of its elements. A group G is called n-centralizer if | Cent(G) |= n, and primitive n-centralizer if | Cent(G) |=| Cent( G Z(G) ) |= n. In this paper, among other things, we investigate the structure of finite groups of odd order with | Cent(G) |= 9 and prove that if |G| is odd, then | Cent(G) |= 9 if and

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Given a finite group G, let t(G) be the number of normal subgroups of G and s(G) be the sum of the orders of the normal subgroups of G. The group G is said to be harmonic if H(G) : jGjt(G)=s(G) is an integer. In this paper, all finite groups for which 1 H(G) 2 have been characterized. Harmonic groups of order pq and of order pqr, where p < q < r are primes, are also classified. Moreover, it has been shown that if G is harmonic and G T C 6 , then t(G) ! 6.