A Characterization of 4-Centralizer Groups

A finite or infinite group is called an -centralizer group if it has numbers of distinct centralizers. In this paper, we prove that a finite or infinite group is a 4-centralizer group if and only if is isomorphic to . This extends a result of Belcastro and Sherman. 1. Introduction Given a finite or infinite group and , the set is called the centralizer of in . The set of all centralizers in is denoted by . A group is called an -centralizer group if . It is easy to see that one-centralizer groups are precisely the abelian groups. Characterization of finite groups in terms of the number of distinct centralizers has been an interesting topic of research in recent years (see [1–8]). In [7], Belcastro and Sherman have proved the nonexistence of finite -centralizer group for . However, their proof also shows the nonexistence of infinite -centralizer groups. Belcastro and Sherman [7] also characterize all finite 4-centralizer groups. More precisely, they proved that a finite group is a 4-centralizer group if and only if is isomorphic to , where is the center of and is the cyclic group having two elements. In this paper, we extend the same characterization for infinite groups using elementary techniques of group theory. Throughout this paper will denote a finite or infinite group. Recall that for any group , its center is the intersection of all centralizers in . Also is the union of all the centralizers of noncentral elements of . It may be mentioned here that a finite or infinite group can not be written as union of two of its proper subgroups. These facts have important role in proving the main theorem of this paper. 2. Main Result In this section, we proof the following main theorem of this paper. Theorem 1. A finite or infinite group is a 4-centralizer group if and only if . Proof. Let be a 4-centralizer finite or infinite group and , where , , and are noncentral elements of . Then , since is the union of its proper centralizers. Let us consider the centralizer . Then will be one of , , , or . If , then . This implies that for some . Therefore, we get , a contradiction. If , then gives Therefore, . Hence, , a contradiction, as a group can not be written as union of two of its proper subgroups. Similarly, it can be seen that . Thus, and so . In a similar way it can be seen that and so . We will now show that . Clearly, . Let . Then Therefore, and so . Thus, In a similar way, it can be seen that , and . Let us consider the right cosets , , , and , where . As is a noncentral element of it follows that , , and . If , then we have for some . Therefore, we get ,