# The Structure of EAP-Groups and Self-Autopermutable Subgroups

Authors: Shima Housieni; Mohammad Reza Rajabzadeh Moghaddam
Journal: The Scientific World Journal
 (2014)

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## Abstract

A subgroupH of a given group G is said to be autopermutable,
if HHα=HαH for all α∈AutG. We also
call H a self-autopermutable subgroup of G, when
HHα=HαH implies that Hα=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.


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## Body

## 1. Introduction

LetH be a subgroup of a given group G. Then we call H to be autopermutable, if HHα=HαH for all α∈Aut(G). The subgroup H is said to be self-autopermutable, if HHα=HαH implies that Hα=H. Moreover, we call the group G to be an EAP-group if every subgroup of G is autopermutable. Clearly, if α 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 H=b of the Dihedral group D8=a,b:a4=b2=1,ab=a-1 is conjugate-permutable, which is not autopermutable. To see this, consider the automorphism α which sends a and b into a and ab, respectively. Clearly HHα is not a subgroup of G, which means that HHα≠HαH. It is easily seen that similar examples can be obtained by taking a direct product of D8 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, D8 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 index4. 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 index4. In fact we prove the following theorem.Theorem 1.
LetG be a finite group with the centre of index 4. Then G is an EAP-group if and only if the Sylow 2-subgroup of G is one of the following forms: (i)
Q8;(ii)
a,b∣a2n+1=b2=1,ab=a2n+1, n≥3;(iii)
Z2×Q8;(iv)
a,b,c,d∣a2=b2=c4=d4=1,b∈ZG,a,c=a,d=c3d,c,d=1;(v)
a,b,c,d,e∣a2=b2=c2=d2=e4=1,a,b,c∈ZG,d,e=b;(vi)
a,b,c,d,e∣a2=b2=c2=d4=e4=1,a,b,c∈ZG,d,e=a.We remind that a nonabelian group is said to be Hamiltonian, if all of its subgroups are normal. The following result gives our claim, whenG is a 2-group with cyclic centre of index 4.Theorem 2.
LetG be a finite 2-group with cyclic centre of index 4. Then G is an EAP-group if and only if G≅Q8 or a,b∣a2n+1=b2=1,ab=a2n+1, for all n≥3.Proof.
Consider the groupG to be Q8. Since Q8 is Hamiltonian group, the result follows easily. Now assume G=a,b∣a2n+1=b2=1,ab=a2n+1, n≥3. One can easily check that G contains exactly three proper subgroups of orders 2i, for 1≤i≤n+1. We also observe that the subgroups of orders 2 are autopermutable and as the subgroups of orders 2n+1 are normal, they are also autopermutable. Now, one can check that there are exactly two cyclic and one noncyclic subgroups of orders 2i, 2≤i≤n, so that one of the cyclic subgroups is central and hence all the subgroups of G satisfy the required property.
Conversely, assume thatG is an EAP-group, Z(G)=x∣x2n=1, G/Z(G)={Z(G),aZ(G),bZ(G),abZ(G)}, where a2,b2∈Z(G) and so a,b≤2n+1. In case n=1, then the group G is either D8 or Q8. As explained before, D8 cannot be an EAP-group and hence G≅Q8. Now suppose n>1 and the elements a and b are both of order 2. Then every element y∈G has the following form (as G is nilpotent of class 2):
(1)y=aibjxk,where0≤i,j<2,0≤k<2n.
Clearly, the map α given by α(y)=biajxk is an automorphism of G, which sends a into b. Thus HHα≠HαH for the subgroup H=〈b〉, which contradicts the assumption. Now, if |a|,|b|<2n+1 we may replace a and b by the elements axi and bxj, both of which are of order 2. This reduces to the previous case. Therefore we must have a or b of order 2n+1. Then G has a cyclic subgroup of order 2n+1 and so G is of order 2n+2(n>1) with the centre of index 4. Hence, by [4, 5.3.4], the group G has the following presentation:
(2)G=a,b∣a2n+1=b2=1,ab=a2n+1,n≥3.
This is an EAP-group and so the proof is completed.The following result considers the case when
G
is a 2-group with noncyclic centre of index 4.Theorem 3.
LetG be a finite 2-group with noncyclic centre of index 4. Then G is an EAP-group if and only if G is one of the following forms:(i)
G=Z2×Q8;(ii)
G=a,b,c,d∣a2=b2=c4=d4=1,b∈ZG,a,c=a,d=c3d,c,d=1;(iii)
G=a,b,c,d,e∣a2=b2=c2=d2=e4=1,a,b,c∈ZG,d,e=b;(iv)
G=a,b,c,d,e∣a2=b2=c2=d4=e4=1,a,b,c∈ZG,d,e=a.Proof.
The sufficient condition is obvious. We only need to prove the necessity condition. LetG be an EAP-group and G/Z(G)={Z(G),aZ(G),bZ(G),abZ(G)}, where a2,b2∈Z(G). Assume that Z(G) is not an elementary abelian 2-group. Since Z(G) 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 Z(G) is an elementary abelian 2-group. Clearly G must be a group of order either 16 or 32. The structure of such groups is given as follows in [5]. If G=16, then (i)
G=Z2×D8;(ii)
G=a,b∣a4=b4=1,bab-1=a3≅Z4⋊Z4;(iii)
G=Z2×Q8;(iv)
G=a,b,c,d∣a2=b2=c4=d4=1,b∈ZG,a,c=a,d=c3d,c,d=1.
AsD8 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 H=〈b〉 and α∈Aut(G) which sends a and b into a and ab, respectively. Clearly, HHα≠HαH and hence G cannot be an EAP-group. Thus when |a|=|b|, then G is of the form given in either (iii) or (iv).
AssumeG=32. Then such groups in the list of small groups with elementary abelian centres of index 4 are only of the following forms: (i)
G=a,b,c∣a4=b4=c2=1,a,b=c,a,c=b,c=1;(ii)
G=a,b,c,d,e∣a2=b2=c2=d4=e4=1,a,b,c∈ZG,d,e=b;(iii)
G=a,b,c,d,e∣a2=b2=c2=d2=e2=1,c,d∈ZG,a,e=b,e=ab,a,b=1;(iv)
G=a,b,c,d,e∣a2=b2=c2=d2=e4=1,a,b,c∈ZG,d,e=b;(v)
G=a,b,c,d,e∣a2=b2=c2=d4=e4=1,a,b,c∈ZG,d,e=a.
For the group of form (i) we may consider the cyclic subgroupH=〈b〉 and α∈Aut(G), which sends a, b, and c into a, ab, and c, respectively. In case the group G is of form (ii), we consider H=〈e〉 and α∈Aut(G) which sends a, b, c, d, and e into a, ab, ac, be, and ed, respectively. Also if the group is considered to be of form (iii), one may consider H=〈e〉 and α∈Aut(G) which sends a, b, c, d, and e into e, b, c, d, and a, respectively. Now, one can easily check that in these cases HHα≠HαH and so G cannot be an EAP-group. Hence, when |G|=32, then G is of either form (iv) or form (v). The proof is complete.Proof of Theorem1.
The necessity condition is obvious and Theorems2 and 3 establish the result, when G is a 2-group. If G is not a 2-group, then we may write G=S1×S2⋯×Sk, in such a way that S1 is a Sylow 2-subgroup and Si is an abelian Sylow pi-subgroup, where pi is an odd prime number, for 2≤i≤k. Clearly, Aut(G)≅Aut(S1)×Aut(S2)×⋯×Aut(Sk) and for any subgroup H of G, H≅H1×H2×⋯×Hk, where Hi≤Si for 1≤i≤k. Thus H is an autopermutable subgroup of G if H1 is an autopermutable subgroup of S1. This completes the proof.


## 3. Self-Autopermutable Subgroups in Nilpotent Groups

We call a subgroupH of a given group G to beweakly characteristic, when Hα≤NG(H) implies that Hα=H for all α∈Aut(G). Also, given the subgroups H and K, then H satisfies thesubcharacteriser condition, if H⊴K implies that NAut(G)(K)≤NAut(G)(H), where NAut(G)(K)={α∈Aut(G);Kα=K}≤Aut(G). Clearly, if one considers the inner automorphisms of the group thenweakly normal andnormaliser condition properties are obtained.The following result of [6] shows that self-conjugate-permutability, weakly normal property, and subnormaliser condition are equivalent for p-subgroups of a given group.Theorem 4 (see [6], Proposition 3.3).
LetH be a p-subgroup of a group G. Then the following properties are equivalent: (i)
H is a self-conjugate-permutable subgroup;(ii)
H is a weakly normal subgroup;(iii)
H satisfies the subnormaliser condition.In this section, it is shown that self-autopermutable subgroups in nilpotent groups are always characteristic.Proposition 5.
LetH be a subgroup of a group G. (i)
IfH is self-autopermutable, then H is weakly characteristic in G.(ii)
IfH is weakly characteristic, then H satisfies the subcharacteriser condition in G.Proof.
(i) IfHα≤NG(H), as H⊴NG(H), we have HHα=HαH. Applying the condition that H is self-autopermutable subgroup of the group G, we get Hα=H. By definition, H is weakly characteristic.
(ii) LetK≤G, such that H⊴K. We have Hα≤Kα=K≤NG(H) for every α∈NAut(G)(K). Since H is weakly characteristic in G, we have Hα=H. Thus α∈NAut(G)(H) and the result is obtained.The following theorem is one of the main results in this section.Theorem 6.
LetH be a subgroup of a nilpotent finite group G. If H satisfies the subcharacteriser condition then H is characteristic in G.Proof.
WriteG≅P1×P2×⋯×Pt, where Pi is a Sylow pi-subgroup of G, for 1≤i≤t. We may also write H≅H1×H2×⋯×Ht, with Hi≤Pi, 1≤i≤t. Since H satisfies the subcharacteriser condition in G, one can easily see that Hi satisfies the subcharacteriser condition in Pi. Therefore Hi⊴Pi implies that Hi is characteristic in Pi, 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.
LetH be a subgroup of a finite nilpotent group G. Then (i)
H is a self-autopermutable;(ii)
H is a weakly characteristic;(iii)
H satisfies the subcharacteriser condition in G.Proof.
The result follows by Proposition5 and Theorem 6.



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*Source: 850526-2014-12-11.xml*