The Structure of EAP-Groups and Self-Autopermutable Subgroups

A subgroup H of a given group G is said to be autopermutable, if HH α = H α H for all α ∈ Aut(G). We also call H a self-autopermutable subgroup of G, when HH α = H α H implies that H α = H. 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.


Introduction
Let be a subgroup of a given group . Then we call to be autopermutable, if = for all ∈ Aut( ). 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 conjugatepermutability [1], self-conjugate-permutability [2], and ECPgroups [3], respectively. One notes that the subgroup = ⟨ ⟩ of the Dihedral group 8 = ⟨ , : 4 = 2 = 1, = −1 ⟩ 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 8 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 selfautopermutable. Moreover, 8 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 4. We also prove that self-autopermutability and characteristic properties are equivalent in nilpotent groups.

Finite EAP-Groups
In this section, we determine the structure of finite EAPgroups, when their centres are of index 4. In fact we prove the following theorem. (ii) ⟨ , | 2 +1 = 2 = 1, = 2 +1 ⟩, ≥ 3; (iii) Z 2 × 8 ; 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 4.
Proof. Consider the group to be 8 . Since 8 is Hamiltonian group, the result follows easily. Now assume = ⟨ , | 2 +1 = 2 = 1, = 2 +1 ⟩, ≥ 3. One can easily check that contains exactly three proper subgroups of orders 2 , for 1 ≤ ≤ + 1. We also observe that the subgroups of orders 2 are autopermutable and as the subgroups of orders 2 +1 are normal, they are also autopermutable. Now, one can check that there are exactly two cyclic and one noncyclic subgroups of orders 2 , 2 ≤ ≤ , 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 2 , 2 ∈ ( ) and so | |, | | ≤ 2 +1 . In case = 1, then the group is either 8 or 8 . As explained before, 8 cannot be an EAP-group and hence ≅ 8 . Now suppose > 1 and the elements and are both of order 2. Then every element ∈ has the following form (as is nilpotent of class 2): Clearly, the map given by ( ) = is an automorphism of , which sends into . Thus ̸ = for the subgroup = ⟨ ⟩, which contradicts the assumption. Now, if | |, | | < 2 +1 we may replace and by the elements and , both of which are of order 2. This reduces to the previous case. Therefore we must have or of order 2 +1 . Then has a cyclic subgroup of order 2 +1 and so is of order 2 +2 ( > 1) with the centre of index 4. 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. Proof. The sufficient condition is obvious. We only need to prove the necessity condition. Let be an EAP-group and / ( ) = { ( ), ( ), ( ), ( )}, where 2 , 2 ∈ ( ). Assume that ( ) is not an elementary abelian 2group. Since ( ) is the direct product of its cyclic subgroups, by the same argument as in Theorem 2, there are no EAPgroups in this case. Now, assume that ( ) is an elementary abelian 2-group. Clearly must be a group of order either 16 or 32. The structure of such groups is given as follows in [5]. If | | = 16, then As 8 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 ∈ Aut( ) 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 | | = 32. Then such groups in the list of small groups with elementary abelian centres of index 4 are only of the following forms: For the group of form (i) we may consider the cyclic subgroup = ⟨ ⟩ and ∈ Aut( ), which sends , , and into , , and , respectively. In case the group is of form (ii), we consider = ⟨ ⟩ and ∈ Aut( ) which sends , , , , and into , , , , and , respectively. Also if the group is considered to be of form (iii), one may consider = ⟨ ⟩ and ∈ Aut( ) which sends , , , , and into , , , , and , respectively. Now, one can easily check that in these cases ̸ = and so cannot be an EAP-group. Hence, when | | = 32, 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 = 1 × 2 ⋅ ⋅ ⋅ × , The Scientific World Journal 3 in such a way that 1 is a Sylow 2-subgroup and is an abelian Sylow -subgroup, where is an odd prime number, for 2 ≤ ≤ . Clearly, Aut( ) ≅ Aut( 1 ) × Aut( 2 ) × ⋅ ⋅ ⋅ × Aut( ) and for any subgroup of , ≤ for 1 ≤ ≤ . Thus is an autopermutable subgroup of if 1 is an autopermutable subgroup of 1 . This completes the proof.

Self-Autopermutable Subgroups in Nilpotent Groups
We call a subgroup of a given group to be weakly characteristic, when ≤ ( ) implies that = for all ∈ Aut( ). Also, given the subgroups and , then satisfies the subcharacteriser condition, if 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-conjugatepermutability, 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: (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 .
Applying the condition that is self-autopermutable subgroup of the group , we get = . By definition, is weakly characteristic.
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 ≅ 1 × 2 × ⋅ ⋅ ⋅ × , where is a Sylow -subgroup of , for 1 ≤ ≤ . We may also write ≅ 1 × 2 × ⋅ ⋅ ⋅ × , with ≤ , 1 ≤ ≤ . 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.