A sequence of factorizable subgroups

Let G be a non-abelian non-simple group. In this article the group G such that G=MCG(M) will be studied, where M is a proper maximal subgroup of G and CG(M) is the centralizer of M in G.

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Дата:	2010
Автор:	Dabbaghian, V.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2010
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/154864
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	A sequence of factorizable subgroups / V. Dabbaghian// Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 19–28. — Бібліогр.: 2 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1548642019-06-17T01:31:22Z A sequence of factorizable subgroups Dabbaghian, V. Let G be a non-abelian non-simple group. In this article the group G such that G=MCG(M) will be studied, where M is a proper maximal subgroup of G and CG(M) is the centralizer of M in G. 2010 Article A sequence of factorizable subgroups / V. Dabbaghian// Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 19–28. — Бібліогр.: 2 назв. — англ. 2000 Mathematics Subject Classification:20E28; 20F14. http://dspace.nbuv.gov.ua/handle/123456789/154864 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Let G be a non-abelian non-simple group. In this article the group G such that G=MCG(M) will be studied, where M is a proper maximal subgroup of G and CG(M) is the centralizer of M in G.
format	Article
author	Dabbaghian, V.
spellingShingle	Dabbaghian, V. A sequence of factorizable subgroups Algebra and Discrete Mathematics
author_facet	Dabbaghian, V.
author_sort	Dabbaghian, V.
title	A sequence of factorizable subgroups
title_short	A sequence of factorizable subgroups
title_full	A sequence of factorizable subgroups
title_fullStr	A sequence of factorizable subgroups
title_full_unstemmed	A sequence of factorizable subgroups
title_sort	sequence of factorizable subgroups
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/154864
citation_txt	A sequence of factorizable subgroups / V. Dabbaghian// Algebra and Discrete Mathematics. — 2010. — Vol. 10, № 2. — С. 19–28. — Бібліогр.: 2 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT dabbaghianv asequenceoffactorizablesubgroups AT dabbaghianv sequenceoffactorizablesubgroups
first_indexed	2025-07-14T06:56:03Z
last_indexed	2025-07-14T06:56:03Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 10 (2010). Number 2. pp. 19 – 28 c© Journal “Algebra and Discrete Mathematics” A sequence of factorizable subgroups Vahid Dabbaghian Communicated by V. I. Sushchansky Abstract. Let G be a non-abelian non-simple group. In this article the group G such that G = MCG(M) will be studied, where M is a proper maximal subgroup of G and CG(M) is the centralizer of M in G. 1. Introduction Let G be a group, and let M and N be two subgroups of G. The group G is called central factorizable if G can be written as the central product of the subgroups M and N . In this case we say M and N are CF-subgroups of G (Central Factorizer subgroup), and we have G/M ∩N ∼= G/M ⊕G/N. (1) Since M ⊆ CG(N) and N ⊆ CG(M), so G = MCG(M) = NCG(N) are the other representations of the central factorizability of G. Therefore M is a CF-subgroup of G whenever G = MCG(M). One notes that every CF-subgroup is normal, hence simple groups are the first example of groups without any proper CF-subgroups. Clearly every subgroup of an abelian group is a CF-subgroup. We are interested to the case that M and CG(M) are proper subgroups. Thus if M is a proper maximal subgroup of G such that Z(G) 6⊂ M , then M is a CF-subgroup (CF-maximal subgroup). Indeed, if Z(G) 6⊂ Φ(G), the Frattini subgroup of G, then G contains a CF-maximal subgroup. 2000 Mathematics Subject Classification: 20E28; 20F14. Key words and phrases: central product, maximal subgroup, sequence of sub- groups. 20 A sequence of factorizable subgroups Definition 1. Let S = {Gn} be a sequence of subgroups of G, indexed by the non negative integers. We call S a CF-sequence of G if 1. G0 = G, 2. Gn = GmZ(Gn) for all m > n and 3. Gn+1 is a proper maximal subgroup of Gn. According to this definition, Gn is a non-abelian non-simple group for every n, and Gm is a CF-subgroup of Gn for all m > n. Let n be a positive integer and n = pα1 1 . . . pαt t be the prime decompo- sition of n. Define Ω(n) = t∑ i=1 αi, Ω(1) = 0 and Ω(∞) = ∞. Let D(G) = Z(G) ∩ Φ(G) and ΩG = Ω([Z(G) : D(G)]), where [Z(G) : D(G)] denotes the index of D(G) in Z(G). We prove the following theorem. Theorem 1. If G is a group and Φ(H) ⊂ Φ(G) for every normal subgroup H with finite index, then G has a CF-sequence of length ΩG. In the final section we extend this work to abstract classes of groups by defining two closure operations C0 and C, finite central product and (infinite) central product, respectively. In particular we prove, Theorem 2. Let A be the class of abelian groups, and X and Y two C0-closed classes of groups such that A ≤ X. Let G be a group and M a CF-maximal subgroup of G. 1. If M is an X-group then so is G. 2. If Y is H-closed and M is an XY-group then G is an XY-group. 2. Upper central series of a CF-sequence Let S = {Gn} be a CF-sequence of a group G. In this section we study the upper central series of the terms of S and we extend it to their lower central and derived series. Lemma 1. Let M be a CF-subgroup of G and C = CG(M). Then 1. G/Z(G) ∼= M/Z(M)⊕ C/Z(C), V. Dabbaghian 21 2. G/Z(M) ∼= G/M ⊕M/Z(M) and 3. C = Z(G) if M is maximal. Proof. Since M and C are CF-subgroups, Z(M) ⊆ Z(G) and Z(C) ⊆ Z(G). Thus Z(M) = M ∩ Z(G) and Z(C) = C ∩ Z(G) = Z(G). Using equation (1) and M ∩ C = Z(M) we have G/Z(G) ∼= MZ(G)/Z(G)⊕ C/Z(G) ∼= M/Z(M)⊕ C/Z(C), G/Z(M) ∼= M/Z(M)⊕ C/Z(M) and G/M ∼= C/Z(M). Hence G/Z(M) ∼= G/M ⊕M/Z(M). Since M ⊆ CG(C), if M is maximal then CG(C) = M or CG(C) = G. If M = CG(C) then Z(M) = M ∩ C = CG(C) ∩ C = Z(C) and G/M ∼= C/Z(C). This implies the index of Z(C) in C is a prime and C is abelian, which is a contradiction. Hence CG(C) = G and C = Z(G). From the part (3) of Lemma 1 we have G/Z(G) ∼= M/Z(M), (2) G/M ∼= Z(G)/Z(M) (3) and G/Z(M) ∼= M/Z(M)⊕ Z(G)/Z(M). (4) Also M is a CF-maximal subgroup of G if and only if Z(G) 6⊂ M . Therefore, the necessary and sufficient condition for G to contain a CF-maximal subgroup is Z(G) 6⊂ Φ(G). Thus when the centre of G is trivial, G has no CF-maximal subgroups. The following proposition is a generalization of Lemma 1 for terms of the upper central series of members of a CF-sequence {Gn}. For simplicity we denote Zn = Z(Gn) and Zα,n = Zα(Gn), the α-th term of the upper central series of Gn for each α and n. Proposition 1. Let {Gn} be a CF-sequence of G. Then for every m and n that m > n, and each α, we have 1. Gm ∩ Zα,n = Zα,m, 2. Gn/Gm ∼= Zα,n/Zα,m, 22 A sequence of factorizable subgroups 3. Gn/Zα,n ∼= Gm/Zα,m and 4. Gn/Zα,m ∼= Gm/Zα,m ⊕ Zn/Zm. Proof. Let m = n+ k. We prove it by induction on k and α. Let α = 1. If k = 1 then Gn+1 is a CF-maximal subgroup of Gn and equations (2), (3) and (4) result it. Suppose the proposition is correct for k − 1, then Zn+k−1 = Gn+k−1 ∩ Zn and Zn+k−1 ⊆ Zn. Since Zn+k ⊆ Zn+k−1 so Zn+k ⊆ Gn+k−1 ∩ Zn. By assumption Gn = Gn+kZn, thus Gn+k ∩ Zn ⊆ Gn+k ∩ CGn(Gnk ) = Zn+k and we get Gn+k ∩ Zn = Zn+k, which is Gm ∩ Zn = Zm. (5) Hence Zn/Zm = Zn/Gm ∩ Zn ∼= GmZn/Gm = Gn/Gm, and Gm/Zm = Gm/Gm ∩ Zn ∼= GmZn/Zn = Gn/Zn. Using equations (1) and (5) we have Gn/Zm ∼= Gm/Zm ⊕ Zn/Zm. This completes the induction on k. Let the above conclusions be correct for α − 1 and Gn/Zα−1,n ∼= Gm/Zα−1,m. Since the groups of inner automorphisms of two isomorphic groups are isomorphic, so (3) as required. Now we show Gm ∩ Zα,n = Zα,m. Since {Gn} is a CF-sequence, we have Gn = GmZn = GmZα,n, thus Gn/Zα,n ∼= Gm/Gm ∩ Zα,n, and using part (3) Gm/Zα,m ∼= Gm/Gm ∩ Zα,n. Therefore, it is enough to show Gm ∩ Zα,n ⊆ Zα,m or (Gm ∩ Zα,n)/Zα−1,m ⊆ Z(Gm/Zα−1,m) = Zα,m/Zα−1,m. Let xZα−1,m ∈ (Gm ∩ Zα,n)/Zα−1,m and yZα−1,m ∈ Gm/Zα−1,m, where x ∈ Gm ∩ Zα,n, y ∈ Gm and x, y /∈ Zα−1,m. Since x ∈ Gm and Gm ∩ Zα−1,n = Zα−1,m, (6) V. Dabbaghian 23 we have x /∈ Zα−1,n and xZα−1,n ∈ Zα,n/Zα−1,n = Z(Gn/Zα−1,n). Also from y ∈ Gm ⊆ Gn and equation (6) we have y /∈ Zα−1,n and yZα−1,n ∈ Gn/Zα−1,n. This proves xyZα−1,n = yxZα−1,n. If xyZα−1,m 6= yxZα−1,m then xyx−1y−1 /∈ Zα−1,m. Since xyx−1y−1 ∈ Gm so xyx−1y−1 /∈ Zα−1,n, which is a contradiction. Thus Gm ∩ Zα,n ⊆ Zα,m and (1) as required. Using part (1) Zα,n/Zα,m = Zα,n/Gm ∩ Zα,n ∼= GmZα,n/Gm = Gn/Gm, which results part (2). Finally by equation (1) and Gm ∩ Zn ⊆ Zα,m = Gm ∩ Zα,n we get Gn/Zα,m = Gm/Zα,m ⊕ ZnZα,m/Zα,m ∼= Gm/Zα,m ⊕ Zn/Zm. This implies part (4). If G = MN is a central factorizable group, then it is easy to prove Zk(G) = Zk(M)Zk(N), γk(G) = γk(M)γk(N), and G(k) = M (k)N (k), where Zk(G), γk(G) and G(k) are k-th term of the upper, lower and derived series of G, respectively. Hence if {Gn} is a CF-sequence of G then for each m > n, 1. Zk(Gn) = Zk(Gm)Z(G) when k ≥ 1, 2. G (k) n = G (k) m when k ≥ 1 and 3. γk(Gn) = γk(Gm) when k ≥ 2. In particular, if G is nilpotent then cl(G) = cl(Gn), and if G is soluble then d(G) = d(Gn) for each n, where cl(G) and d(G) are nilpotency class and defect of a given group G, respectively. 24 A sequence of factorizable subgroups 3. Groups with a CF-sequence Proof of Theorem 1. Case I) ΩG = 0. In this case Z(G) ⊆ Φ(G) and G has no CF-maximal subgroup. Thus G has no CF-sequences. Case II) ΩG = ∞. Let G0 = G. Then Z(G0) has infinite order. Since ΩG0 = ∞,Z(G0) 6⊆ Φ(G0). In this case there exist a CF-maximal subgroup G1 of G0 such that G0 = G1Z(G0) and G0/G1 ∼= Z(G0)/Z(G1). Hence [Z(G0) : Z(G1)] is prime and Z(G1) has infinite order. Since G1 is normal in G0 and has a finite index, Φ(G1) ⊆ Φ(G0). Thus ΩG1 = ∞ and Z(G1) 6⊆ Φ(G1). So there exists a CF-maximal subgroup G2 of G1 such that G1 = G2Z(G1), [G1 : G2] = [Z(G1) : Z(G2)] is prime, and Z(G2) has infinite order. If G0 = G1Z(G) = G2Z(G1)Z(G) then G0 = G2Z(G), G2 is normal in G0 and [G0 : G2] = [Z(G0) : Z(G1)][Z(G1) : Z(G2)] < ∞. Hence Φ(G2) ⊆ Φ(G0) and ΩG2 = ∞. This procedure gives an infinite sequence {Gn} of subgroups of G such that G0 = G and Gn = Gn−1Z(Gn). (7) We show Gn = GmZ(Gn) for m > n. Let m = n + k. We prove it by induction on k. For k = 1 it is (7). Let Gn = Gn+k−1Z(Gn). Since Gn+k−1 = Gn+kZ(Gn+k−1), Gn = Gn+kZ(Gn+k−1)Z(Gn) and Z(Gn+k−1) ⊆ Z(Gn). Thus Gn = Gn+kZ(Gn) and the induction is com- pleted. This shows {Gn} is an infinite CF-sequence. Case III) 0 < ΩG < ∞. Let G0 = G. Since ΩG > 0, thus Z(G0) 6⊂ Φ(G0) and there exists a CF-maximal subgroup G1 of G0 such that G0 = G1Z(G0), and [G0 : G1] = [Z(G0) : Z(G1)] is a prime. So Ω(\|Z(G1)\|) = Ω(\|Z(G0)\|)−1. On the other hand G1 is normal in G0. Therefore Φ(G1) ⊆ Φ(G0). If Z(G1) ⊂ Φ(G0) then ΩG0 = [Z(G0) : D(G0)] = 1 and G1 ≤ G0 is a CF-sequence of G. Otherwise, Z(G1) 6⊆ Φ(G1) and there is a CF- maximal subgroup G2 of G1 of prime index such that G1 = G2Z(G1). Hence G0 = G2Z(G0), G2 ⊳ G0, Φ(G2) ⊂ Φ(G0) and Ω(\|Z(G2)\|) = Ω(\|Z(G0)\|) − 2. Since ΩG0 is finite, after a finite steps we obtain a CF- maximal subgroup Gl of Gl−1 of prime index such that Gl−1 = GlZ(Gl−1), Ω(\|Z(Gl)\|) = Ω(\|Z(G0)\|) − l and Z(Gl) ⊆ Φ(G0). Hence ΩG = l is the length of the sequence Gl ≤ Gl−1 ≤ · · · ≤ G1 ≤ G0 = G, where Gn = Gn+1Z(Gn) for 1 ≤ n ≤ ΩG. V. Dabbaghian 25 Now it is easy to see that Gn = GmZ(Gn) for 1 ≤ n < m ≤ ΩG. This completes the proof. Recall a group G is called with max if each non-empty set of subgroups of G has a maximal element. Corollary 1. If G is with max then G has a CF-sequence of length ΩG. Proof. If G is with max then every subgroup of G is finitely generated and Φ(H) ⊆ Φ(G) for every normal subgroup H of G. The most well known classes of groups with max are finitely generated nilpotent groups and polycyclic groups. Therefore, using Theorem 1 and Corollary 1, each finitely generated nilpotent group and polycyclic group has a CF-sequence of length ΩG. As we showed in section 2, when G is a finitely generated group then each term of its CF-sequence is nilpotent of class cl(G). Similarly, when G is a polycyclic group then each term of its CF-sequence is soluble with defect d(G). 4. On abstract classes of groups In this section we generalize some senses of pervious sections and discuss about the invariant properties by the central product. Suppose A and B are two subgroups of a group G such that [A,B] = 1. Then the central product AB is a subgroup of G. It is also correct for ∏ i∈I Ci where {Gi}i∈I is a collection of distinct subgroups of G, such that [Gi, Gj ] = 1 for i 6= j. Our interest is the cases that AB and ∏ i∈I Ci are X-groups, whenever A, B and every Ci are X-groups for an abstract class of groups X. Let X be a class of groups. We say the class X is closed under taking finite central product, or C0-closed, if C1,C2 are the X-subgroups of G such that [G1, G2] = 1 and C1C2 is a X-subgroup of G. Similarly X is C-closed or central product closed, if ∏ i∈I Ci, the (central) product of the subgroups Ci, is an X-group, for a collection {Gi}i∈I of distinct X-subgroups of G. It is easy to see that C0 and C are two closure operations. By definition, the class of abelian groups is C-closed (and so C0-closed). It is proved that if X is a {S,H, P }-closed class of groups and G = AB is a group, which is the product of X-groups A and B, then G is an X-group, whenever one of them, A or B, are subnormal in G [1]. Since in the central product AB both of A and B are normal in G, thus every abstract class of groups X which is {S,H, P }-closed, is C0-closed. 26 A sequence of factorizable subgroups Proposition 2. 1. D0 ≤ C0 ≤ N0. 2. D ≤ C ≤ N . Proof. It is enough to prove (2) then (1) is clear. Let X be an N -closed class of groups and {Gi}i∈I be a collection of distinct X-subgroups Ci of G such that [Ci, Cj ] = 1 for i 6= j. Since Ci is normal in C = ∏ i∈I Ci, thus C ∈ X and this requires X is C-closed. If Ci ∩ Cj = 1 for every i, j ∈ I, which i 6= j, then C = Dri∈ICi. This implies X is D-closed. As a corollary, the class of nilpotent groups is N0-closed, so is C0- closed. Also by [2, page 34], the class of periodic groups is N -closed, which is C-closed. In the following theorem we obtain some new C-closed classes of groups. Theorem 3. Let X and Y be two C-closed abstract classes of groups. 1. If X is S-closed then LλX is C-closed for every cardinal number λ. 2. If Y is H-closed then XY is C-closed. The class LλX is defined to consist of all groups G in which every subset of cardinality at most λ is contained in a X-subgroup of G, for a cardinal number λ. Before the proof of Theorem 3, we refer to this fact that, if H and K are two subgroups of G and N ⊳ G then [HN/N,KN/N ] = [H,K]N/N. (8) Proof of Theorem 3. Let {Gi}i∈I be a collection of distinct subgroups of G such that [Ci, Cj ] = 1 for i, j ∈ I and i 6= j. Let Ci be an LλX-group for each i ∈ I. We show the central product C = ∏ i∈I Ci is also an LλX-group. Suppose X = 〈xk\|k ∈ K〉 is a subgroup of C, generated by elements xk for k ∈ K, where K is an index set with cardinal number at most λ. As xk ∈ C and [Ci, Cj ] = 1 for i 6= j, we can write down xk = ck1 · · · ckt , where cki ∈ Cki and ki 6= kj for 1 ≤ i 6= j ≤ t. Let C∗ ki be the subgroup of Cki generated by all elements cki of Cki which appear in the representation of xk. Then it is clear that for every ki ∈ I, C∗ ki has the cardinal number at most λ and so it is an X-group. But X is C-closed class of groups and [C∗ ki , C∗ kj ] = 1 for ki, kj ∈ I and ki 6= kj . Hence the central product C∗ = ∏ k∈I C ∗ ki is an X-group. Since X is S-closed, it is enough to show X ≤ C∗. Let x be an element of X. Then x is a product of some generated elements xk. Since for every pair elements cki and ckj that ki 6= kj , V. Dabbaghian 27 [cki , ckj ] = 1, thus we can write x as a product of elements cki such that no pair of them belongs to the same Ci. This implies x ∈ C∗. Let Ci ∈ XY for every i ∈ I. Then there exists a normal subgroup Di in Ci such that Di ∈ X and Ci/Di ∈ Y. As X is C-closed and [Di, Dj ] = 1 for i 6= j, the central product D = ∏ i∈I Di is a normal subgroup of C and D ∈ X. Now it is enough to show C/D ∈ Y. We have C/D = ( ∏ i∈I Ci)/D = ∏ i∈I (CiD/D) ∼= ∏ i∈I (Ci/(Ci ∩D)), (9) Ci/(Ci ∩D) ∼= (Ci/Di)/((Ci ∩D)/Di) and Y is H-closed, thus Ci/(Ci ∩D), which is the homomorphic image of Ci/Di, is a Y-group. On the other hand, by equation (8) [CiD/D,CjD/D] = [Ci, Cj ]D/D = 1 for every i, j ∈ I and i 6= j and Y is C-closed. Therefore using (9) we get C/D ∈ Y. This completes the proof. If we substitute C0-closed instead of C-closed in the theorem above then the proof will be correct in a similar way. Corollary 2. If X is a {S,C}-closed ({S0, C0}-closed) class of groups then the class LX is C-closed (C0-closed). Proof. As for every finite cardinal number λ, Lλ ≤ L so as required. The class of locally soluble groups is C0-closed, while it is not N0- closed [2, page 90]. It is easy to see that the classes of FC-groups and CC-groups are C0-closed but they are not N0-closed. Also the class of abelian groups which is C-closed, is not N -closed. The class X of groups of even powers of a prime number p is an example of D0-closed class of groups which is not C0-closed: Let G be an abelian group and A = 〈r〉 ⊕ 〈s〉, B = 〈t〉⊕〈s〉 be two X-subgroups of G such that \|r\| = \|s\| = p3 and \|s\| = p. Then [A,B] = 1 and AB is not an X-group. Hence X is not C0-closed, while it is obviously D0-closed. Proof of Theorem 2. Since M is a CF-maximal X-subgroup of G, so G = MZ(G). By assumption, Z(G) is an X-group and X is C0-closed, hence G is an X-group. This proves (1). Since any given group G is an extension of itself by the trivial group, thus if G ∈ X then G ∈ XY for the class of groups Y. Hence A ≤ XY. On the other hand, Y is H-closed and using Theorem 3, the class XY is C0-closed. Using part (1) if M is a CF-maximal XY-subgroup of G then G is an XY-group. 28 A sequence of factorizable subgroups Acknowledgment This research was supported in part by NSERC discovery grant. References [1] B. Amberg, Infinite factorized groups, In Group-Korea 1988 Proceeding, Lecture Notes in Mathematics 1398, Springer, Berlin. [2] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Vol 1 and 2, Springer-Verlag, Berlin 1972. Contact information V. Dabbaghian MoCSSy Program, The IRMACS Centre, Si- mon Fraser University, Burnaby V5A 1S6, Canada E-Mail: vdabbagh@sfu.ca Received by the editors: 15.12.2009 and in final form 25.02.2011.

A sequence of factorizable subgroups

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