The Structure of EAP-Groups and Self-Autopermutable Subgroups

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.