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Permutability of Subgroups of Semidirect Product of P-Groups 1

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Permutability of Subgroups of Semidirect Product of P-Groups 1 International Journal of Algebra, Vol. 13, 2019, no. 7, 331 - 339 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2019.9828 Permutability of Subgroups of Semidirect Product of p-Groups Bilal N. Al-Hasanat Department of Mathematics Al Hussein Bin Talal University Ma'an, Jordan This article is distributed under the Creative Commons by-nc-nd Attribution License. Copyright c 2019 Hikari Ltd. Abstract Two subgroups H and K of a group G are called permutable if HK = KH. Furthermore, H is permutable subgroup (quasinormal subgroup) if HK = KH for any subgroup K of G. Certainly, every normal subgroup is permutable. Permutability does not imply normality [2]. Permutability of subgroups of G × H that are direct products of subgroups of the direct factors were investigated in [6]. In this article, the classification of all permutable subgroups of the group G = Zpk oZp will be investigated for any odd prime p and for any integer k ≥ 2. On the other hand, many properties of the group G were studied and configured. Mathematics Subject Classification: 22E46, 53C35, 57S20 Keywords: Permutable subgroup, normal subgroup, group centre, gener- ator, semidirect product, capable group 1 Introduction The classification of all 2-generator p-group of nilpotency class 2 was given in [1]. The authors classified these groups using the fact that they are all central extensions of a cyclic p-group by an abelian p-group of rank 2. ∼ ∼ For groups G = Zn = hai and H = Zm = hbi, the semidirect product G o H is completely determined by how a acts on b. In this case, the action is 332 Bilal N. Al-Hasanat −1 given by conjugation. So, the action of a on b is b ab. Clearly, hai C G o H. Thus, this action is a power of a. That is b−1ab = ai, which indicates that [a; b] = ai−1, where ai and n are coprime, and im ≡ 1( mod m). In this article, we give a complete classification of the group G = Zpk o Zp (for p odd prime and 2 ≤ k 2 N). Significantly, these groups have always p subgroups which are permutable and not normal. In addition to the other distinctive properties the group has. All of these properties concerning group G will be introduced and studied. In [2], the authors determined all permutable subgroups of groups of order 16, and determined which subgroup is permutable and not normal. In this article, we will extend this result, and describe all subgroups of the group G = Zpk o Zp (for p odd prime and 2 ≤ k 2 N), in which permutability does not coincide with normality. 2 Notation and general setup The cyclic group of order n will be denoted by Zn. The identity element of a group G is e. The order of an element x 2 G is the least positive integer k such that xk = e and denoted by o(x) = k. The cyclic subgroup generated by x is hxi = fxk j k = 1; 2; ··· ; o(x)g. The centre of a group G is denoted by Z(G), and Z(G) = fx 2 G j xy = yx ; for all y 2 Gg. A subgroup H of a group G is called normal subgroup of G (H E G) if and only if xH = Hx for all x 2 G. Clearly, Z(G) E G. This implies that, if x 2 Z(G), then hxi E G. The next results will be useful in our investigation. Theorem 2.1. [7](Cauchy's theorem) Given a finite group G and a prime number p dividing the order of G, then there exists an element (and hence a cyclic subgroup) of order p in G. Theorem 2.2. [7] [First Sylow theorem] For every prime factor p with multi- plicity n of the order of a finite group G, there exists a Sylow p-subgroup of G of order pn. Lemma 2.1. [4] If G is a group and N is a normal subgroup of G, then G=N is abelian if and only if N contains the commutator subgroup of G. 3 Main results The main aim of this article is to classifying the subgroups structure of a group G = Zpk o Zp. These groups have certain properties, in which there were some subgroups which permutable and not normal, and all other subgroups are Permutability of subgroups of semidirect product of p-groups 333 normal. On the other hand, this article will discuss some of main topics in group theory for the groups Zpk o Zp, where p is an odd prime and k is an integer with k ≥ 2. D E 0 pk−1 ∼ Corollary 3.1. The derived subgroup of G = Zpk o Zp is G = a = Zp 0 and G E G. k Proof. Let G = Zpk o Zp. Then G has two generators a and b of orders p and p, respectively. The commutator of a and b is [a; b] = a−1b−1ab = a−pk−1 , D E D E which implies that apk−1 = a−pk−1 ⊆ G0. Note that jG0j = p and since D E D E pk−1 0 pk−1 0 pk−1 ∼ o(a ) = p, then G ⊆ a . Thus G = a = Zp. For Z(G) E G and G=Z(G) is abelian. Using Lemma 2.1, implies that Z(G) contains G0. Hence, 0 G E G, and the claim. Remark 3.1. The lower central series of a group G is: G = γ0(G) ≥ γ1(G) ≥ · · · ≥ γc(G) ≥ · · · where γi(G) = [γi−1(G);G] for i = 1; 2; 3; ··· . The group G is called nilpotent if there exists c such that γc(G) = feg, and the smallest such c is the class of nilpotency. Corollary 3.2. The group G = Zpk o Zp is nilpotent group of class 2, and solvable. Proof. The group G is a 2-generator p-group. Let G = ha; bi where o(a) = pk and o(b) = p. Note that, [x; y] = ampk−1 , for m 2 f1; 2; ··· ; pg and pk−1 0 D pk−1 E a 2 Z(G) (Corollary 3.1). Then, γ1(G) = [G; G] = G = a , and so 0 γ2(G) = [γ1(G);G] = [G ;G] = feg. Hence, the nilpotency class of G is c = 2. Which also implies that, G is solvable. The next theorem has been proved in [1] is used here to describe all finite 2-generator p-group of nilpotency class 2. Theorem 3.1. [1] Let G be a finite 2-generator p-group of class 2 (p an odd prime). Consider the integers α; β; γ and σ with (α ≥ β ≥ γ ≥ σ ≥ 0). Then G is isomorphic to exactly one group of the following three types: D E ∼ pα pβ 1. G = G1 = a; b j a = b = [a; b; a] = [a; b; b] = e where α; β; γ are integers and γ ≥ 1. D E ∼ pα pβ pα−γ 2. G = G2 = a; b j a = b = [a; b; a] = [a; b; b] = e; a = [a; b] where α ≥ 2γ and γ ≥ 1. 334 Bilal N. Al-Hasanat D E ∼ pα pβ pα+σ−γ pσ 3. G = G3 = a; b j a = b = [a; b; a] = [a; b; b] = e; a = [a; b] , where α + σ ≥ 2γ and γ > σ ≥ 1. The next lemma gives the orders of the groups listed in the previous theo- rem. Lemma 3.1. [1] Let G be a 2-generator p-group of class 2 (p an odd prime). ∼ α+β+γ 1. If G = G1, then jGj = p . ∼ α+β 2. If G = G2, then jGj = p . ∼ α+β+σ 3. If G = G3, then jGj = p . Using Theorem 3.1 and the above lemma, we give the representation of G = Zpk o Zp in the following theorem. Theorem 3.2. For any odd prime p and any integer k ≥ 2, the group G = Zpk o Zp is non-abelian 2-generators p-group, and it can be represented by D k k−1 E G = a; b j ap = bp = e ; ap = [b; a] (1) Proof. From the previous results, the group G = Zpk o Zp is a 2-generator p-group of class 2. Therefore, G is isomorphic to one of the groups listed in Theorem 3.1, by setting α = k ; β = 1 and γ = 1. Since, [a; b] = a−1b−1ab, so [a; b; a] = (ab)−1b(ba)−1aba = (ba1−pk−1 )−1ba−pk−1 b−1a−1aba = (ba1−pk−1 )−1(ba1−pk−1 ) = e. Similarly for [a; b; b] = e. Also, jGj = o(a)o(b) = pk+1 = pα+β, and since k ≥ 2 then α ≥ 2γ. This implies that, Equation 1 satisfies exactly the axioms of the group G2 in Theorem 3.1. Moreover, jG0j = p1 = pγ. Corollary 3.3. Let G = Zpk o Zp, for an odd prime p and an integer k ≥ 2. p ∼ Then, the centre of G, Z(G) = ha i = Zpk−1 . Proof. Lemma 2.1 shows that Z(G) is non-trivial, and since G is non-abelian group, then p ≤ jZ(G)j < pk+1, and so G=Z(G) is abelian group and not cyclic. ∼ k−1 Thus, G=Z(G) = Zp × Zp, implies that jZ(G)j = p . For any x 2 G and since [x; a] 2 Z(G), one has, (xa)p = xpap[x; a−1]p(p−1)=2 = xpap [x; a−1]p(p−1)=2 = xpap This implies that the pth map φ : G ! G, defined by φ(x) = xp, is a homo- morphism.
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