We consider sufficient conditions for metacyclic groups to split. Specifically, we show that a finite metacyclic group G of odd order is split on its cyclic normal subgroup K if K is such that G/K is cyclic and |K| = exp G.Розглянуто достатнi умови для розщеплення метациклiчних груп, а саме, показано, що скiнченна метациклiчна група G непарного порядку розщеплюється на своїй циклiчнiй нормальнiй пiдгрупi K, якщо K є такою, що G/K є циклiчною та |K| = exp G

He-guo Liu

Yan Yang

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 512.5Yan Yang, He-guo Liu (Hubei Univ., Wuhan; Hubei Univ. Arts and Sci., Xiangyang, China)ON SPLIT METACYCLIC GROUPS *ПРО РОЗЩЕПЛЮВАНI МЕТАЦИКЛIЧНI ГРУПИWe consider sufficient conditions for metacyclic groups to split. Speciifically, we show that a finite metacyclic group ofodd order G is split on its cyclic normal subgroup K if K is such that G/K is cyclic and |K| = expG.Розглянуто достатнi умови для розщеплення метациклiчних груп, а саме, показано, що скiнченна метациклiчнагрупа G непарного порядку розщеплюється на своїй циклiчнiй нормальнiй пiдгрупi K, якщо K є такою, що G/Kє циклiчною та |K| = expG.1. Introduction. G is a metacyclic group if and only if there exists a cyclic normal subgroup K ofG such that G/K is cyclic, and G = SK is called a metacyclic factorization, when S is cyclic. Inparticular, if G has a split metacyclic factorization G = SK such that S ∩K = 1, then G is calledsplit metacyclic group, otherwise is called nonsplit. C. E. Hempel and Hyo-Seob Sim have given theclassifications of metacyclic groups in their papers [1, 2].Resently, the structure of the automorphism group of metacyclic group have been given muchattention. The automorphism groups of the split metacyclic p-groups have been given in [3] for p isodd and in [4] for p = 2. And the automorphism groups of the finite split metacyclic groups havebeen given in [5]. But the case of nonsplit groups is much more complicated than the case of splitgroup, and only the automorphism groups of nonsplit metacyclic p-group of odd order have beengiven in [6]. Thus it is necessary to determine wether the metacyclic group is split or not, before westudy its automorphism. And in this paper, of particular interest are some sufficient coditions to showa special type of finite metacyclic groups are split.2. Notation and preliminaries. In this section, we present some general facts that will be usefulin this paper. We first define some notation which will be kept throughout.pi(G): the set of all prime divisors of the order of a finite group G.r(p): the largest integer i such that pi divides the positive integer r.G is a metacyclic group if and only if there exists a cyclic normal subgroup K of G such thatG/K is cyclic. And such a subgroup K is called a kernel of G.Lemma 1. A p-group P of odd order is metacyclic with a kernel of order pγ and of index pαif and only if it has a presentationP = 〈a, b | apα = bpβ , bpγ = 1, ba = bpδ+1〉,where α, β, γ, δ are positive integers such that γ ≤ min(α+ δ, β + δ).Lemma 2 ([1], Lemma 2.1). A group G is metacyclic with a kernel of order γ and of index α ifand only if it has a presentationG = 〈a, b | aα = bβ, bγ = 1, ba = bδ〉,where α, β, γ, δ are positive integers such that γ|δα − 1 and γ|β(δ − 1).*This work was supported by the Natural Science Foundation of Hubei Univ. Arts and Sci. (Grant No.2010YA002).c© YAN YANG, HE-GUO LIU, 20121432 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10ON SPLIT METACYCLIC GROUPS 1433Lemma 3 ([2], Lemma 3.4). Let P = SK be a metacyclic factorization of a p-group P of oddorder. P is present by the presentationP = 〈x, y |xpα = ypβ , ypγ = 1, yx = ypδ+1〉,wherepα = |S : S ∩K|, pβ = |K : S ∩K|, pγ = |K|, pδ = |K : P ′|.Lemma 4 ([5], Corollary 4.1). Let p be a odd prime, k, s, t be non-negative integers. Then(i) (1 + pk)pst ≡ 1 (mod pk+s);(ii) if k > 0, then1 + rs + . . .+ rs(pt−1) ≡ pt (mod pt+k),for r = 1 + pk.Lemma 5 ([2], Lemma 5.3). Let G be a metacyclic group with a metacyclic factorization G == SK. To each set pi of prime numbers, the subgroup H = SpiKpi is the unique Hall pi-subgroup ofG such that Spi = S ∩H and Kpi = K ∩H, so H = (S ∩H)(K ∩H).Definition 1 ([2], Definition 5.4). Let G be a group with a metacyclic factorization G = SKand let pi denote the set {p ∈ pi(G) : G has a normal Hall p′-subgroup}. Let H denote the Hallpi-subgroup SpiKpi and let N denote the the Hall pi′-subgroup Spi′Kpi′ . Then the semidirect decompo-sition G = HN is called the standard Hall-decomposition for the metacyclic factorization G = SK.Lemma 6 ([2], Lemma 5.6). Let G = HN be a Hall-decomposition for the metacyclic factor-ization G = SK and pi={p ∈ pi(G) : G has a normal Hall p′-subgroup}. Then(i) H = SpiKpi = (S ∩H)(K ∩H), and H is nilpotent;(ii) N = Spi′Kpi′ = (S ∩N)(K ∩N), and Kpi′ = G′ ∩N, Spi′ ∩Kpi′ = 1;(iii) Spi′ ≤ CN (H).3. Split metacyclic group. 3.1. Split metacyclic p-group.Lemma 7. If P = SK be a metacyclic factorization of a p-group P of odd order, thenexpP = max (|S|, |K|).Proof. By Lemma 3, P can be represented asP = 〈a, b|apα = bpβ , bpγ = 1, ba = bpδ+1〉,where |a| = |S| and |b| = |K|. For any asbt ∈ P, by Lemma 4, we have(asbt)pm= aspmbt(1+(pδ+1)s+...+(pδ+1)s(pm−1)) = aspmbt(pm+kpδ+m) = 1,where pm = max (|S|, |K|) and k is an integer.Lemma 8. P is a metacyclic p-group of odd order. If K is a normal cyclic subgroup with|K| = expP, then P/K is cyclic.Proof. LetP = 〈a, b|apα = bpβ , bpγ = 1, ba = bpδ+1〉,be a presentation of P. Suppose K = 〈ambn〉, and we observe that |K| = expP = max(|a|, |b|)from Lemma 7. Then by Lemma 4 we haveISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 101434 YAN YANG, HE-GUO LIU(apbp)pγ−1= apγbp(1+(1+pδ)p+...+(1+pδ)p(pγ−1−1)) = 1,which shows (m, p) = 1 or (n, p) = 1. If (m, p) = 1, we know that P/K is cyclic since P == 〈b〉〈ambn〉. If (n, p) = 1, P/K is also cyclic since P = 〈a〉〈ambn〉.Theorem 1. Let P be a metacyclic p-group of odd order. If P has a normal cyclic subgroup Kof order expP, then P is split.Proof. P/K is cyclic by Lemma 8. Let P = HK be a metacyclic factorization of P. Thus byLemma 3, choosing the generators x and y for H and K, respectively, we have the presentationP = 〈x, y |xpα = ypβ , ypγ = 1, yx = ypδ+1〉,wherepα = |H : H ∩K| = |P |/ expP, pβ = |K : H ∩K|,pγ = |K| = expP, pδ = |K : P ′| = |P/P ′|/pα.If β = γ, then P is split. Now we assume that β < γ. Denote a = xyfpβ−α, where f = pγ−β − 1,then we will show 〈a〉 ∩ 〈y〉 = 1. From Lemma 1 we can obtainapα= (xyfpβ−α)pα= xpαyfpβ−α(1+(pδ+1)+...+(pδ+1)pα−1) = xpαyfpβ−αpα == ypβ+fpβ = ypγ= 1,andapα−1= (xyfpβ−α)pα−1= xpα−1yfpβ−αpα−1 /∈ K,which implies 〈a〉 ∩ 〈y〉 = 1. P = 〈a〉〈y〉 is obvious. Thus P is split. Finally, because ofya = yxyfpβ−α= yx = y1+pδ,we can get the presentation of PP = 〈a, y | apα = ypγ = 1, ya = ypδ+1〉,where pα = |P |/ expP, pγ = expP, pα+δ = |P/P ′|.Example 1. The above theorem may not hold for a metacyclic 2-groupQ8 = 〈x, a |x2 = a2, a4 = 1, ax = a−1〉,where |a| = 4 = expQ8 and 〈a〉 C Q8, but Q8 is not a split group.Applying Theorem 1 and Lemma 1, we can deduce the following corollary.Corollary 1. A metacyclic p-group P of odd order is non-split if and only if P has the presen-tation of the form in Lemma 1P = 〈a, b | apα = bpβ , bpγ = 1, ba = bpδ+1〉,where α > β > δ ≥ 1 and β < γ ≤ β + δ.ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10ON SPLIT METACYCLIC GROUPS 14353.2. Split metacyclic group. In the remaining part of this section, we will proof that the conclusionabove is also hold for a metacyclic group of odd order.Lemma 9. Let P be a non-abelian metacyclic p-group of odd order. Suppose P = SK andP = S1K1 both are the split metacyclic factorizations. Then(i) |S| = |S1| and |K| = |K1|;(ii) P = SK1 and P = S1K also are the split metacyclic factorizations. Here, P = SK is asplit metacyclic factorization means P = SK is a metacyclic factorization and S ∩K = 1.Proof. (i) SupposeP = SK = 〈a, b | apα = bpγ = 1, ba = bpδ+1〉,P = S1K1 = 〈a1, b1 | apα11 = bpγ11 = 1, ba11 = bpδ1+11 〉.It is obvious that |P | = pα+γ = pα1+γ1 and expP = max (α, γ) = max (α1, γ1). Thus if α = α1,(i) holds by Lemma 7. If α 6= α1, then α = γ1 and α1 = γ. And we observe that P/P ′ ∼=∼= Zpα × Zpδ ∼= Zpα1 × Zpδ1 . Consequently,γ1 = α = δ1, γ = α1 = δ,which contradicts that P is non-abelian. Thus α = α1, γ = γ1, δ = δ1.(ii) Let b1 = ambn, where m,n are non-negative integers. We will show that (n, p) = 1. Supposep|n. Thenbpδ1 = (ambn)pδ= bp(1+(1+pδ)m+...+(1+pδ)m(pδ−1)) ∈ 〈bpδ+1〉,which contradicts that P ′ = 〈bpδ〉 = 〈bpδ1 〉 = 〈(ambn)pδ〉, since |K1/P ′| = |K/P ′| = pδ. Therefore,P = 〈a, b〉 = 〈a, b1〉 = SK1 and S ∩K1 = 1.Theorem 2. Let G be a metacyclic group of odd order and Y is a normal cyclic subgroup ofG with |Y | = expG. Then G/Y is cyclic and G is split .Proof. Let G = HN be a Hall-decomposition for a metacyclic factorization G = SK, pi == pi(H)={p ∈ pi(G) : G has a normal Hall p′-subgroup} and pi′ = pi(N). Thus Spi = S ∩ H,Kpi = K ∩H,Spi′ = S ∩N, Kpi′ = K ∩N, by Lemma 5.For p ∈ pi, denote Hp and Yp = 〈yp〉 as the sylow p-subgroup of H and Y, respectively. ThusYp  Hp and |Yp| = expHp. Then from Lemma 9, we know Hp is split. Applying Lemma 9, wecan find a xp ∈ Hp, such that Hp = 〈xp〉〈yp〉 is a split metacyclic factorization. Let x1 =∏p∈pi xp,y1 =∏p∈pi yp. Then H = 〈x1〉〈y1〉 is a split metacyclic factorization, since H is nilpotent.If N = 1, then G = H is split. If N is non-trivial, then from Lemma 7, we know N = Spi′Kpi′is a split metacyclic factorazition, since Spi′ ∩ Kpi′ = 1. Next we will show N = Spi′Ypi′ is a splitmetacyclic factorization, where Ypi′ is the Hall pi′-subgroup of Y.For any q ∈ pi′, denote Yq and Nq = SqKq as the the sylow q-subgroup of Y and N, respectively.Note that Yq  Nq and |Yq| = expNq. We know Yq is a kernel of Nq by Theorem 1. This impliesthat Nq = SqYq is a split metacyclic factorization by Sq ∩Kq = 1 and Lemma 9. ThusN =∏q∈pi(N)Nq =∏q∈pi(N)SqKq =∏q∈pi(N)SqYq =∏q∈pi(N)Sq∏q∈pi(N)Yq = Spi′Ypi′ .ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 101436 YAN YANG, HE-GUO LIUFuther, Spi′ ∩ Ypi′ = 1, since Sq ∩ Yq = 1.Let Spi′ = 〈x2〉 and Ypi′ = 〈y2〉, then Spi′ ≤ CN (H) and Y G yields thatG = HN = 〈x1〉〈y1〉〈x2〉〈y2〉 = 〈x1〉〈x2〉〈y1〉〈y2〉 = 〈x1x2〉〈y1y2〉 = XY,where X = 〈x1x2〉 and X ∩ Y = 1.The above theorem may not hold for a group of even order.Example 2. LetG ∼= Q8 × Z3 = 〈x, b |x2 = b6, b12 = 1, bx = b7〉,where |b| = 12 = expG and 〈b〉G, but G is not a split group.In Theorem 2, if Y is not normal, G may not be a split group.Example 3. LetG = 〈a, b | a34 = b33 , b35 = 1, ba = b1+32〉.It is obviously that |a| = expG, but[b, a] = b−1ba = b32 ∈ 〈a〉shows 〈a〉 is not normal. And G is non-split.Lemma 10. If G = SK is a metacyclic factorization of a metacyclic group G of odd order.Then expG = lcm (|S|, |K|).Proof. Let S = 〈x〉,K = 〈y〉, we will show that|xmyn| ≤ lcm (|S|, |K|) ∀xmyn ∈ G.Let r = lcm(|S|, |K|) and t = |K|. Suppose yx = yθ, and it is obvious that θr ≡ 1(mod |K|).Then for p ∈ pi(G) and kpr(p) = rθr = θkpr(p) ≡ 1 (mod pt(p)),which implies θmkpr(p) ≡ 1(mod pt(p)). Then from Lemma 4 we have1 + θmk + . . .+ θmk(pr(p)−1) ≡ pr(p)(mod pt(p)).Thus1 + θm + . . . θm(r−1) =k−1∑i=0θmi(1 + θmk + . . .+ θmk(pr(p)−1)) =⇒=⇒ pt(p)|1 + θm + . . .+ θm(r−1) =⇒ t|1 + θm + . . .+ θm(r−1) =⇒=⇒ (xmyn)r = xmryn(1+θm+...+θm(r−1)) = 1.Corollary 2. Let G = SK be a metacyclic factorization of a metacyclic group G of odd order,and |K|∣∣|S|. Then G is split.ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10ON SPLIT METACYCLIC GROUPS 1437Example 4. Let G = SK = 〈a〉〈b〉 be a metacyclic factorization.(i) Suppose G = 〈a, b | a32 = b33 , b35 = 1, ba = b1+33〉. Then |K| = 35 = expG yields that G issplit. Let x = ab24, and we can get a presentation of GG = 〈x, b |x32 = 1, b35 = 1, ba = b1+33〉.(ii) Suppose G = 〈a, b | a34 = b33 , b35 = 1, ba = b1+33〉. Then S  G and |S| = 36 = expGyields that G is split, for P ′ = 〈b33〉 S. Let x = b−1a6, and we can get a presentation of GG = 〈x, a |x33 = 1, a36 = 1, ax = b1+34〉.(iii) Suppose G = 〈a, b | a33 = b33×7, b34×7 = 1, ba = b415〉. Then |K| = 34 × 7 = expG yieldsthat G is split. Let x = ab14, and we can get a presentation of GG = 〈x, b |x33 = 1, b34×7 = 1, bx = b415〉.1. Hempel C. E. Metacyclic groups // Communs Algebra. – 2000. – 28. – P. 3865 – 3897.2. Hyo-Seob Sim. Metacyclic groups of odd order // Proc. London Math. Soc. – 1994. – 69. – P. 47 – 71.3. Bidwell J. N. S, Curran M. J. The automorphism group of a split metacyclic p-group // Arch. Math. – 2006. – 87. –P. 488 – 497.4. Curran M. J. The automorphism group of a split metacyclic 2-group // Arch. Math. – 2007. – 89. – P. 10 – 23.5. Golasinski M., Goncalves D. L. On automorphisms of split metacyclic groups // Manuscr. Math. – 2009. – 128. –P. 251 – 273.6. Curran M. J. The automorphism group of a nonsplit metacyclic p-group // Arch. Math. – 2008. – 90. – P. 483 – 489.Received 03.04.12ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10

On split metacyclic groups

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On split metacyclic groups

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)