Let be an automorphism of a given group ; then, is called the autocentralizer of in . In this work, we study finite groups , which can be written as the union of autocentralizers of some automorphisms of . In particular, if a group is 5--centralizers, then we determine the absolute central quotient of , where . Finally, it is shown that autocommutative property of a group is equivalent to the one, in which every -centralizer of its nontrivial elements is abelian.

1. Introduction

Let be the group of all automorphisms of a given group and . The autocentralizer or A-centralizer of in is the subgroup of consisting of all elements of which are fixed by and denoted by , i.e.,

Also, we denote the set of all such autocentralizes in by . Note that, if , where is the identity element of , then is called fixed-point-free automorphism. We remark that, for the identity automorphism . of , we have , and hence, .

For any element and , the element,is the autocommutator of and . Clearly, if is the inner automorphism induced by the element of , then the autocommutator coincides with the ordinary commutator . The set , consisting of all elements in which are fixed by every automorphism of , is a central characteristic subgroup of , which is called absolute centre or the autocentre of (see [15], for more information). Hence, we may write

We denote the number of distinct autocentralizers of automorphisms in by , and now a question arises that “how many distinct autocentralizers can a group have?”

This article is motivated by providing some answers to the above question.

2. Preliminary Results

It is clear that, in a given group , there may exist an element , which is not fixed by none of nontrivial automorphisms of . For example, consider cyclic groups of order , for odd prime .

In the present article, we consider all finite groups which do not have the above property, so the following result is obvious.

Lemma 1. A group is the union of autocentralizers of all nontrivial automorphisms of in , i.e., .

Proof. Clearly, . Now, if , then, by the above assumption, there exists , such that . Therefore, , which completes the proof.

Remark 1. Note that if , then (3) implies that , so , for all and . Hence, , and consequently, or . Clearly, the converse is also true. Therefore, if and only if or .

Lemma 2. Let be a nontrivial group except ; then, .

Proof. By Remark 1, . If , then is the proper subgroup of itself, which is impossible. Suppose that ; then, , for some automorphisms and , where and are proper subgroups of . Therefore, , which is impossible, since a group cannot be written as the union of two proper subgroups. Thus, .

A group is called --centralizer if (see also [6]).

Example 1. (i)Consider , the Dihedral group of order 8; then, clearly, the set of all automorphisms of isThus, one can calculate thatTherefore, is 4--centralizer, as and are not distinct.Similarly, one can easily see that .(ii)Consider , the symmetric group of order 6; then, one can calculate that has 5 -centralizers as follows:

3. Counting -Centalizers in Groups

For an arbitrary element of a given group , the set , where is the inner automorphism of , is called the centralizer of in . The set of all such centralizers in is denoted by . Clearly, is abelian if and only if .

Many authors have studied the influence of the number of centralizers of a finite group on the structure of the group. In 1994, Belcastro and Sherman [7] proved that , for any nonabelian finite group . They also showed that has 4 centralizers if and only if and has 5 centralizers if and only if or . In [8], Ashrafi proved that if has 6 centralizers, then or , where is the alternating group of degree 4.

In this section, we compute under certain conditions on the group .

Proposition 1. Let be the Dihedral group of order ; then,

Proof. Clearly, for , the group has the following presentation:Then, , so . Assume that is odd; then, we have the following -centralizers:(i), as , , and , for every .(ii), as , , and , for every and even .(iii), as , for every , , and , when .By similar argument, one has , so if is odd.Now, assume that is even; then, we have the following cases(iv), as , , and , for every .(v), as , when , , and , when .Using similar argument for , when is even, we have the following -centralizers , so if is even.

The following result of [9] is useful in our further investigations.

Theorem 1 (see Theorem 3.1 in [9]). Let be a group with , for any prime number . Then, is isomorphic to one of the following groups:(1)(2)(3)(4)(5)(6)

Using the above theorem, we have the following.

Theorem 2. Let be a group such that , for any prime number . Then, .

Proof. Let and ; then, Proposition 1 implies that . Clearly, , and it is easily checked that .
Assume ; then, clearly,Hence, one can calculate that the group has the following -centralizers:In the cases and , example 1 and Proposition 1 show that is 4--centralizer.
If , then with relatively simple and long calculations, we obtain that has the following -centralizers:Now, let and be an odd prime number. Then, it is clear that or , for every . Thus, we have the following -centralizers:where , so has distinct -centralizers.

The following theorem of [10] is needed to prove our next result.

Theorem 3 (see Theorem 1 in [10]). A group is the nontrivial union of three subgroups if and only if it is homomorphic to the Klein four-group.

Proposition 2. Let be a group; then, if and only if .

Proof. Using Theorem 1, it is enough to show that , which implies .
Suppose ; then, , for some . Hence, by Lemma 1, . Consider , which will be one of the -centralizers , or . If , then , for all , so , which is a contradiction.
Assume , then for any , we have . Therefore , and hence, , which is again a contradiction. Similarly, if , then we have a contradiction, so must be equal with .
Now, Theorem 3 implies that is isomorphic with Klein four-group.

Theorem 4. Let be a finite group with ; then, or .

Proof. Assume and is -centralizer, for some automorphisms of . Hence, Lemma 1 implies that .
Now, consider , which should be one of , or . Thus, we have the following cases:(a)If , then , for all , so , which is a contradiction.(b)If , then, for all , we have . Therefore, , and hence, and gives a contradiction.Similarly, if , then we obtain a contradiction, so can be equal to either or .(i)Assume that , and are the -centralizers of . Then, using similar argument as in parts (a) and (b), we have and . On the contrary, . Thus, for all , we haveand hence, . Also, and imply that . Therefore, and .(ii)Assume that , and are the -centralizers of . Then, similar argument as in part (i) implies that .Hence, , for , when .
Now, for computing the value of , we show that if and are arbitrary distinct proper -centralizers of , for ; then,Clearly, . As , we have . Therefore, implies that . On the contrary, one observes thatand hence, . Assume ; then, is cyclic and Theorem 2.2 of [9] implies that or , which both give contradictions as has fixed-point-free automorphism, and does not conform to the conditions given at the beginning of the second section. Therefore, .
Now, without loss of generality, we may assume that . Suppose ; then, we havewhich is a contradiction. Hence, or . If , we obtainOne can easily calculate thatso .
Now, applying (3) to and , we have , and hence, . That is, , so or . The property implies that . Therefore, the value of must be one of , or .
Now, if , then divides , and hence, is impossible. Similarly, . Hence, we have the following cases:If , then Theorem 3.5 of [9] and example 1 (iii) imply that and . Again Theorem 2.2 of [9] implies that if , then , which does not conform to the conditions given at the beginning of Section 2.
Let ; then, as divides if , then , and if , then , which both give contradictions. Therefore, . On the contrary, the property implies that , and hence, . As , we obtain , which is again a contradiction. So, cannot be equal to .
Finally, assume that and divides . If , then , and if , then , which are both impossible. Therefore, . Now, again implies that . Also, implies that , which is a contradiction. Hence, .
Now, assume that . In this case, usingwe have . Thus, . On the contrary, , so . Therefore, or . Again applying on and , we obtainThus, , and hence, , or .
Assume that , and as divides , we must have , which is impossible. Similarly, , and . Also, assume that , , and or ; then, againimplies that or , respectively, which are both impossible. Hence, , so we have one of the following cases:or

4. Groups with Abelian -Centralizers

The concept of commutative transitive groups was first introduced and studied by Weisner [11] in 1925.

In this section, we introduce the new concept of autocommutative transitive groups, which is a generalized version of commutative transitive groups. Also, we study a group , in which every -centralizer of a nontrivial element of is abelian. We show that such groups are equivalent to autocommutative transitive groups.

Definition 1. A group is autocommutative transitive if and imply that , for any nontrivial elements in and every .

If runs over the inner automorphisms of , then one has the usual commutative transitive groups.

Lemma 3. For any group , the following statements are equivalent:(i) is A-CT group(ii)The -centralizers of nontrivial automorphisms of are abelian

Proof. Let be A-CT group. For any nonidentity automorphism element , if , we have and . The definition of A-CT implies that . Hence, is abelian. Assume are nontrivial elements of , with and , for every . Obviously, , by the assumption is abelian, and hence, . Thus, is autocommutative transitive.

Using the above lemma, we have the following.

Corollary 1. Let be a finite A-CT group and be a set of pairwise noncommuting elements of with maximal size. Then, .

Data Availability

The datasets used and analysed during the study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.