Abstract

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 , a contradiction. Again, if , then which gives , a contradiction. Similarly, it can be seen that . Thus the cosets , , , and are mutually disjoint. Clearly, .
Let then . Suppose that then without any loss of generality we may assume that . Let us consider the centralizer . Then is one of , , , or .
Case 1. Let . In this case, and so which gives . Therefore, , a contradiction.
Case 2. Let . In this case, which gives . Therefore, , a contradiction.
Case 3. Let . In this case, which gives . Therefore, , a contradiction.
Hence, , since is a 4-centralizer group. Now, implies that which gives since . Thus, . Also, implies that . Hence, and so for some . In other words, . Therefore, . Hence, . This shows that . Finally, the fact that is nonabelian gives .
Conversely, let . Therefore, there are four right cosets of in , namely, , , , and , where , , and are distinct noncentral elements of . So, for any element , either or or or . If , then . If , then for some , therefore . Similarly, gives and gives . Hence, has at most four centralizers, namely, , , , and . Since we have . This completes the proof.

We conclude this paper by the following remark. In [7], Belcastro and Sherman proved Theorem 1 for finite groups only. Note that their proof can not be extended to infinite groups as they have used the finiteness of the group extensively. Belcastro and Sherman [7] have also obtained the structure of 5-centralizer finite groups as follows.

A finite group is a 5-centralizer group if and only if or , where is the cyclic group having three elements and the symmetric group on three symbols.

Note that the if part of the above result also holds for infinite groups. The same proof of Belcastro and Sherman [7] holds for infinite groups. However, the proof of the only if part of the above result is not known for infinite groups. It may be interesting to study the infinite -centralizer groups for .

Acknowledgments

This paper is a part of the author’s M. Phil. thesis done under the supervision of Professor A. K. Das. The author would like to thank him for his helpful suggestions. The author is grateful to all the referees for their valuable comments and suggestions.