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The Scientific World Journal
Volume 2014, Article ID 850526, 3 pages
http://dx.doi.org/10.1155/2014/850526
Research Article

The Structure of EAP-Groups and Self-Autopermutable Subgroups

1Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad 9187147578, Iran
2Department of Mathematics, Khayyam University, Mashhad 9189747178, Iran
3Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Iran

Received 19 May 2014; Accepted 16 September 2014; Published 11 December 2014

Academic Editor: K. C. Sivakumar

Copyright © 2014 Shima Housieni and Mohammad Reza Rajabzadeh Moghaddam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A subgroup H of a given group G is said to be autopermutable, if for all . We also call H a self-autopermutable subgroup of G, when implies that . Moreover, G is said to be EAP-group, if every subgroup of G is autopermutable. One notes that if α runs over the inner automorphisms of the group, one obtains the notions of conjugate-permutability, self-conjugate-permutability, and ECP-groups, which were studied by Foguel in 1997, Li and Meng in 2007, and Xu and Zhang in 2005, respectively. In the present paper, we determine the structure of a finite EAP-group when its centre is of index 4 in G. We also show that self-autopermutability and characteristic properties are equivalent for nilpotent groups.

1. Introduction

Let be a subgroup of a given group . Then we call to be autopermutable, if for all . The subgroup is said to be self-autopermutable, if implies that . Moreover, we call the group to be an EAP-group if every subgroup of is autopermutable. Clearly, if runs over the inner automorphisms of the group, we obtain the notions of conjugate-permutability [1], self-conjugate-permutability [2], and ECP-groups [3], respectively. One notes that the subgroup of the Dihedral group is conjugate-permutable, which is not autopermutable. To see this, consider the automorphism which sends and into and , respectively. Clearly is not a subgroup of , which means that . It is easily seen that similar examples can be obtained by taking a direct product of with any other group. Also, every noncharacteristic normal subgroup of a given group is an example for a self-conjugate-permutable subgroup which is not self-autopermutable. Moreover, is an ECP-group, which is not an EAP-group.

In the present paper, we determine the structure of a finite EAP-group, when its centre is of index . We also prove that self-autopermutability and characteristic properties are equivalent in nilpotent groups.

2. Finite EAP-Groups

In this section, we determine the structure of finite EAP-groups, when their centres are of index . In fact we prove the following theorem.

Theorem 1. Let be a finite group with the centre of index . Then is an EAP-group if and only if the Sylow 2-subgroup of is one of the following forms: (i);(ii), ;(iii);(iv);(v);(vi).

We remind that a nonabelian group is said to be Hamiltonian, if all of its subgroups are normal. The following result gives our claim, when is a 2-group with cyclic centre of index .

Theorem 2. Let be a finite 2-group with cyclic centre of index . Then is an EAP-group if and only if or , for all .

Proof. Consider the group to be . Since is Hamiltonian group, the result follows easily. Now assume , . One can easily check that contains exactly three proper subgroups of orders , for . We also observe that the subgroups of orders 2 are autopermutable and as the subgroups of orders are normal, they are also autopermutable. Now, one can check that there are exactly two cyclic and one noncyclic subgroups of orders , , so that one of the cyclic subgroups is central and hence all the subgroups of satisfy the required property.
Conversely, assume that is an EAP-group, , , where and so . In case , then the group is either or . As explained before, cannot be an EAP-group and hence . Now suppose and the elements and are both of order . Then every element has the following form (as is nilpotent of class ): Clearly, the map given by is an automorphism of , which sends into . Thus for the subgroup , which contradicts the assumption. Now, if we may replace and by the elements and , both of which are of order . This reduces to the previous case. Therefore we must have or of order . Then has a cyclic subgroup of order and so is of order with the centre of index . Hence, by [4, 5.3.4], the group has the following presentation: This is an EAP-group and so the proof is completed.

The following result considers the case when is a 2-group with noncyclic centre of index 4.

Theorem 3. Let be a finite 2-group with noncyclic centre of index . Then is an EAP-group if and only if G is one of the following forms:(i);(ii);(iii);(iv).

Proof. The sufficient condition is obvious. We only need to prove the necessity condition. Let be an EAP-group and , where . Assume that is not an elementary abelian 2-group. Since is the direct product of its cyclic subgroups, by the same argument as in Theorem 2, there are no EAP-groups in this case. Now, assume that is an elementary abelian 2-group. Clearly must be a group of order either or . The structure of such groups is given as follows in [5]. If , then (i);(ii);(iii);(iv).
As is not an EAP-group, hence the group of form (i) cannot be an EAP-group. For the group of form (ii) we can consider and which sends and into and , respectively. Clearly, and hence cannot be an EAP-group. Thus when , then is of the form given in either (iii) or (iv).
Assume . Then such groups in the list of small groups with elementary abelian centres of index are only of the following forms: (i);(ii);(iii);(iv);(v).
For the group of form (i) we may consider the cyclic subgroup and , which sends , , and into , , and , respectively. In case the group is of form (ii), we consider and which sends , , , , and into , , , , and , respectively. Also if the group is considered to be of form (iii), one may consider and which sends , , , , and into , , , , and , respectively. Now, one can easily check that in these cases and so cannot be an EAP-group. Hence, when , then is of either form (iv) or form (v). The proof is complete.

Proof of Theorem 1. The necessity condition is obvious and Theorems 2 and 3 establish the result, when is a 2-group. If is not a 2-group, then we may write , in such a way that is a Sylow 2-subgroup and is an abelian Sylow -subgroup, where is an odd prime number, for . Clearly, and for any subgroup of , , where for . Thus is an autopermutable subgroup of if is an autopermutable subgroup of . This completes the proof.

3. Self-Autopermutable Subgroups in Nilpotent Groups

We call a subgroup of a given group to be weakly characteristic, when implies that for all . Also, given the subgroups and , then satisfies the subcharacteriser condition, if implies that , where . Clearly, if one considers the inner automorphisms of the group then weakly normal and normaliser condition properties are obtained.

The following result of [6] shows that self-conjugate-permutability, weakly normal property, and subnormaliser condition are equivalent for -subgroups of a given group.

Theorem 4 (see [6], Proposition 3.3). Let be a -subgroup of a group . Then the following properties are equivalent: (i) is a self-conjugate-permutable subgroup;(ii) is a weakly normal subgroup;(iii) satisfies the subnormaliser condition.

In this section, it is shown that self-autopermutable subgroups in nilpotent groups are always characteristic.

Proposition 5. Let be a subgroup of a group . (i)If is self-autopermutable, then is weakly characteristic in .(ii)If is weakly characteristic, then satisfies the subcharacteriser condition in .

Proof. (i) If , as , we have . Applying the condition that is self-autopermutable subgroup of the group , we get . By definition, is weakly characteristic.
(ii) Let , such that . We have for every . Since is weakly characteristic in , we have . Thus and the result is obtained.

The following theorem is one of the main results in this section.

Theorem 6. Let be a subgroup of a nilpotent finite group . If satisfies the subcharacteriser condition then is characteristic in .

Proof. Write , where is a Sylow -subgroup of , for . We may also write , with , . Since satisfies the subcharacteriser condition in , one can easily see that satisfies the subcharacteriser condition in . Therefore implies that is characteristic in , which proves the result.

Finally, we show that self-autopermutability, weakly characteristic, and subcharacteriser conditions are equivalent, for every subgroup of a nilpotent group.

Corollary 7. Let be a subgroup of a finite nilpotent group . Then (i) is a self-autopermutable;(ii) is a weakly characteristic;(iii) satisfies the subcharacteriser condition in .

Proof. The result follows by Proposition 5 and Theorem 6.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

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