On Solvable Groups of Arbitrary Derived Length and Small Commutator Length

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Volume 2011, Article ID 245324,5pages doi:10.1155/2011/245324

Research Article

On Solvable Groups of Arbitrary Derived Length

and Small Commutator Length

Mehri Akhavan-Malayeri

Department of Mathematics, Alzahra University, Vanak, Tehran 19834, Iran

Correspondence should be addressed to Mehri Akhavan-Malayeri,[email protected]

Received 2 April 2011; Revised 11 June 2011; Accepted 10 July 2011

Academic Editor: Aloys Krieg

Copyrightq2011 Mehri Akhavan-Malayeri. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Wreath product constructions has been used to obtain for any positive integer n, solvable groups of derived length n, and commutator length at most equal to 2.

1. Introduction

LetGbe a group andGits commutator subgroup. Denote bycGthe minimal number such that every element ofGcan be expressed as a product of at mostcGcommutators. A group

G is called a c-group ifcGis finite. For any positive integer n, denote by cn the class of

groups with commutator length,cG n.

LetFn,tx1, . . . , xnandMn,tx1, . . . , xnbe, respectively, the free nilpotent group

of rank n and nilpotency class t and the free metabelian nilpotent group of rank n and nilpotency classt. Stroud, in his Ph.D. thesis1in 1966, proved that for allt, every element of the commutator subgroupF_n,t can be expressed as a product ofncommutators. In 1985, Allambergenov and Roman’kov2proved thatcMn,tis preciselyn, provided thatn ≥2,

t ≥ 4, orn ≥ 3, t ≥ 3. In 3, Bavard and Meigniez considered the same problem for the

n-generator free metabelian groupMn. They showed that the minimum number cMnof

commutators required to express an arbitrary element of the derived subgroupM_n satisfies

_n

2

≤cMn≤n, 1.1

wheren/2is the greatest integer part ofn/2.

Since Fn,3 groups are metabelian, the result of Allambergenov and Roman’kov 2

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havecMn n, for alln≥2. These results were extended in4to the larger class of abelian

by nilpotent groups, and it was shown thatcG nifGis anon-abelianfree abelian by nilpotent group of rankn.

In5, we proved that 2 ≤ cW ≤ 3, whereW GC∞ is the wreath product of a

nontrivial groupGwith the infinite cyclic group. Recently in6, we have generalized this result. LetWGHbe the wreath product ofGby an-generator abelian groupH. We have proved that every element ofWis a product of at mostn2 commutators, and every element ofW2is a product of at most 3n4 squares inW.

In the case of a finited-generator solvable groupGof solvability lengthr, Hartley7

proved thatcG ≤ d 2d−1r −1. And in a recent paper, Segal8has proved that in a finited-generator solvable groupG, every element ofGcan be expressed as a product of 72d2₄₆_d_commutators.

The problem remains open for thed-generator solvable group in general. In the section of open problems in the site of Magnus projecthttp://www.grouptheory.org, Kargapolov asks the questionS4as follows:

“Is there a number N Nk, d so that every element of the commutator subgroup of a free solvable group of rankkand solvability lengthd, is a product ofNcommutators?”

The answer is “yes” for free metabelian groups; see 2, and for free solvable groups of solvability length 3, see9.

In 10, we found lower and upper bound for the commutator length of a finitely generated nilpotent by abelian group. We also considered an n-generator solvable group

G such thatG has a nilpotent by abelian normal subgroupK of finite index. If K is an s -generator group, thencG≤ss1/272n2₄₇_n_{. We considered the class of solvable group}

of finite Pr ¨ufer ranks, and we proved that every element of its commutator subgroup is equal to a product of at mostss1/272s2₄₇_s_{. And as a consequence of the above results, we}

proved that ifAis a normal subgroup of a solvable groupGsuch thatG/Ais ad-generator finite group andAhas finite Pr ¨ufer ranks, thencG≤ ss1/272s2_n2 ₄₇_s_n_.

These bounds depend only on the number of generators of the groups.

In11, we considered a solvable group satisfying the maximal condition for normal subgroups. We found an upper bound for the commutator length of this class of groups. The bound depends on the number of generators of the groupG, the solvability length of the group, and the number of generators of the groupGk as aG-subgroup. In particular, if in a finitely generated solvable groupG, each term of the derived series is finitely generated as aG-subgroup, thenGis a c-group. We also gave the precise formulas for expressing every element of the derived group to the product of commutators.

In the present paper, we use wreath product constructions to obtain for any positive integern, solvable groups of derived lengthnand commutator length equal to 1 or 2.

2. Main Results

Notation. Let N be a subgroup of a group G, and x, y ∈ G. Then, xy y−1xy, x, y

x−1_y−1_xy_and_{N, x} _{_{n, x}_:_n_∈_N_}_.

The main results of this paper are as follows.

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Lemma 2.1. LetGbe any solvable group, say of derived lengthn, and letT be a cyclic group. Then,

WGT is a solvable group of derived lengthn1.

Theorem 2.2. For any positive integern, there are solvable groups of derived lengthn, in which every

element ofGis a commutator.

Theorem 2.3. The commutator length of the wreath product of ac1-group by the infinite cyclic group

is at most equal to 2.

In particular, we have the following consequences of these results.

Corollary 2.4. For any positive integer n, there are solvable groups of derived length n, with

commutator length at most equal to 2.

Corollary 2.5. For any positive integern, there aren-generator solvable groups of derived lengthn,

in which every element ofGis a commutator.

3. Proofs

The proof ofLemma 2.1is similar to the proof of Lemma 9.22 in12.

Proof ofLemma 2.1. LetT tandW GT. LetB Dr1≤i≤|T|Gi be the base group ofW.

Then,WBT is the semidirect product ofBbyT, where the action ofTonBis given by

gt

i gi1 ∈Gi1. SinceGis solvable of derived lengthn,Bis also solvable of derived lengthn.

SinceW/Bis abelian,Wis solvable andW≤B. It is clear that for anyi,

g_i−₋1₁gig_i−₋1₁g_it₋₁

gi−1, t

∈W. 3.1

Now, assume thatπdenotes the projection ofBon toGi and letπ π|W. In view of3.1, it is clear thatπis surjective. AndWis a solvable group of derived length at leastn. Since

W ≤ Band Bis solvable, of derived length n,Wis a solvable group of derived lengthn. Therefore,Wis of derived length equal ton1.

The proof ofTheorem 2.2requires the following theorem proved in9.

Theorem 3.1Rhemtulla 9. The commutator length of the wreath product of ac1-group by a

finite cyclic group is again ac1-group.

Now, we turn to the proof ofTheorem 2.2.

Proof ofTheorem 2.2. LetAbe any nontrivial abelian group, and letT tbe a finite cyclic

group. DefineG1A, GnGn−1T. Repeated application ofLemma 2.1shows that for every

positive integern,Gnis a solvable group of derived lengthn. By our assumption,G2AT.

LetBbe the base group ofG2. Then,G2BTand

G₂ BT, BT B, T {b, t;b∈B}. 3.2

This easily follows from the relations b, t−1_b−1t−1_{, t} _and_{b, t}_b_{, t}_bb_{, t} _{for all}

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commutator. Now, for every positive integern≥3, sinceGnGn−1Tand every elements of

G_n₋₁is a commutator, repeated application ofLemma 2.1and Rhemtulla’s result shows that the groupGn, obtained by taking successive wreath product of finite cyclic groups satisfies

the desired property and the proof is complete.

The proof ofTheorem 2.3requires the following lemma proved in5.

Lemma 3.2. Let A be a free abelian group andW AwrC_∞, whereC_∞is the infinite cyclic group,

then W is ac-group and furthermore the commutator length ofWis equal to 1.

Proof ofTheorem 2.3. LetWGT, whereGis ac1-group andT t Z. Then,WBT,

where B Dri∈ZGi, whereGi G. Modulo B, W AZ, whereA G/G. Since A is

isomorphic to F/K for some free groupF andAZis a quotient ofF Z, it is clear that

cAZ ≤ cF Z; hence, by Lemma 3.2, every element ofW/B is a commutator. Now,

BDri∈ZG_i, and sinceG∈c1, every element ofWis a product of two commutators.

Remark 3.3. Rhemtulla had introduced in 9a group which is the wreath product of a c1

-group by the infinite cyclic -group, and it is no longer ac1-group.

Now, we proveCorollary 2.4.

Proof ofCorollary 2.4. LetGn−1be the group defined inTheorem 2.2, and letHGn−1Z. Since

Gn−1is ac1-group of derived lengthn−1, it follows fromLemma 2.1thatHis a solvable group

of derived lengthn. Now, to complete the proof, it is enough to applyTheorem 2.3toH.

Finally, we proveCorollary 2.5.

Proof ofCorollary 2.5. Let A be any non trivial cyclic group, T any nontrivial finite cyclic

group, and Gn the group defined in Theorem 2.2. Then, by Theorem 2.3 13, Gn is a n

generator group.

Acknowledgments

The author thanks the editor of the IJMMS and the referee who have patiently read and verified this paper and also suggested valuable comments. The author also likes to acknowledge the support of Alzahra University.

References

1 P. Stroud, Ph.D. thesis, Cambridge, UK, 1966.

2 Kh. S. Allambergenov and V. A. Roman’kov, “Products of commutators in groups,” Doklady Akademii

Nauk UzSSR, no. 4, pp. 14–15, 1984Russian.

3 C. Bavard and G. Meigniez, “Commutateurs dans les groupes m´etab´eliens,” Indagationes

Mathemati-cae, vol. 3, no. 2, pp. 129–135, 1992.

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7 B. Hartley, “Subgroups of finite index in profinite groups,” Mathematische Zeitschrift, vol. 168, no. 1, pp. 71–76, 1979.

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12 J. S. Rose, A Course on Group Theory, Cambridge University Press, London, UK, 1978.

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