Let be a group, and let |Cent( )| denote the number of distinct centralizers of its elements. A group is called -centralizer if |Cent( )| = . In this paper, we investigate the structure of finite groups of odd order with |Cent( )| = 10 and prove that there is no finite nonabelian group of odd order with |Cent( )| = 10.

In this paper, we propose a more secure version of a Cayley hash function which is based on the linear functions. It is a practical parallelizable hash function.

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In this paper, we prove that a group is nilpotent of class at most two if and only if each of its self-centralizing subgroups is normal. We also show that if all self-centralizing subgroups of a certain group are subnormal then all subgroups are subnormal.

-A subgroup H of a group G is said to be -normal in G if there exists a normal subgroup N of G such that HN=G and H∩N ≤ Core(H) where Core(H) is the largest normal subgroup of G contained in H . In this paper we consider finite p-groups of order at most p 4 where p is a prime and show that all of their subgroups are c-normal. Also we study some classes of finite groups whose all of subgroups are c-normal.