Research Article | Open Access
On Solvable Groups of Arbitrary Derived Length and Small Commutator Length
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.
Let be a group and its commutator subgroup. Denote by the minimal number such that every element of can be expressed as a product of at most commutators. A group is called a c-group if is finite. For any positive integer , denote by the class of groups with commutator length, .
Let and be, respectively, the free nilpotent group of rank and nilpotency class and the free metabelian nilpotent group of rank and nilpotency class . Stroud, in his Ph.D. thesis  in 1966, proved that for all , every element of the commutator subgroup can be expressed as a product of commutators. In 1985, Allambergenov and Roman’kov  proved that is precisely , provided that , , or , . In , Bavard and Meigniez considered the same problem for the -generator free metabelian group . They showed that the minimum number of commutators required to express an arbitrary element of the derived subgroup satisfies where is the greatest integer part of .
Since groups are metabelian, the result of Allambergenov and Roman’kov  shows that for , and in , we considered the remaining case . We have , for all . These results were extended in  to the larger class of abelian by nilpotent groups, and it was shown that if is a (non-abelian) free abelian by nilpotent group of rank .
In , we proved that , where is the wreath product of a nontrivial group with the infinite cyclic group. Recently in , we have generalized this result. Let be the wreath product of by a -generator abelian group . We have proved that every element of is a product of at most commutators, and every element of is a product of at most squares in .
In the case of a finite -generator solvable group of solvability length , Hartley  proved that . And in a recent paper, Segal  has proved that in a finite -generator solvable group , every element of can be expressed as a product of commutators.
The problem remains open for the -generator solvable group in general. In the section of open problems in the site of Magnus project (http://www.grouptheory.org/), Kargapolov asks the question as follows:
“Is there a number so that every element of the commutator subgroup of a free solvable group of rank and solvability length , is a product of commutators?”
In , we found lower and upper bound for the commutator length of a finitely generated nilpotent by abelian group. We also considered an -generator solvable group such that has a nilpotent by abelian normal subgroup of finite index. If is an -generator group, then . We considered the class of solvable group of finite Prüfer rank , and we proved that every element of its commutator subgroup is equal to a product of at most . And as a consequence of the above results, we proved that if is a normal subgroup of a solvable group such that is a -generator finite group and has finite Prüfer rank , then . These bounds depend only on the number of generators of the groups.
In , 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 group , the solvability length of the group, and the number of generators of the group as a -subgroup. In particular, if in a finitely generated solvable group , each term of the derived series is finitely generated as a -subgroup, then is 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 integer , solvable groups of derived length and commutator length equal to 1 or 2.
2. Main Results
Notation 2. Let be a subgroup of a group , and . Then, , and .
The main results of this paper are as follows.
First, we need the following generalization of Lemma 9.22 in .
Lemma 2.1. Let be any solvable group, say of derived length , and let be a cyclic group. Then, is a solvable group of derived length .
Theorem 2.2. For any positive integer , there are solvable groups of derived length , in which every element of is a commutator.
Theorem 2.3. The commutator length of the wreath product of a -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 , there are solvable groups of derived length , with commutator length at most equal to 2.
Corollary 2.5. For any positive integer , there are -generator solvable groups of derived length , in which every element of is a commutator.
Proof of Lemma 2.1. Let and . Let be the base group of . Then, is the semidirect product of by , where the action of on is given by . Since is solvable of derived length , is also solvable of derived length .
Since is abelian, is solvable and . It is clear that for any , Now, assume that denotes the projection of on to and let . In view of (3.1), it is clear that is surjective. And is a solvable group of derived length at least . Since and is solvable, of derived length , is a solvable group of derived length . Therefore, is of derived length equal to .
Theorem 3.1 (Rhemtulla ). The commutator length of the wreath product of a -group by a finite cyclic group is again a -group.
Now, we turn to the proof of Theorem 2.2.
Proof of Theorem 2.2. Let be any nontrivial abelian group, and let be a finite cyclic group. Define . Repeated application of Lemma 2.1 shows that for every positive integer , is a solvable group of derived length . By our assumption, . Let be the base group of . Then, and This easily follows from the relations and for all which hold when is a normal abelian subgroup. Hence, every elements of is a commutator. Now, for every positive integer , since and every elements of is a commutator, repeated application of Lemma 2.1 and Rhemtulla's result shows that the group , obtained by taking successive wreath product of finite cyclic groups satisfies the desired property and the proof is complete.
Lemma 3.2. Let A be a free abelian group and , where is the infinite cyclic group, then is a -group and furthermore the commutator length of is equal to 1.
Proof of Theorem 2.3. Let , where is a -group and . Then, , where , where . Modulo , where . Since is isomorphic to for some free group and is a quotient of , it is clear that ; hence, by Lemma 3.2, every element of is a commutator. Now, , and since , every element of is a product of two commutators.
Remark 3.3. Rhemtulla had introduced in  a group which is the wreath product of a -group by the infinite cyclic group, and it is no longer a -group.
Now, we prove Corollary 2.4.
Proof of Corollary 2.4. Let be the group defined in Theorem 2.2, and let . Since is a -group of derived length , it follows from Lemma 2.1 that is a solvable group of derived length . Now, to complete the proof, it is enough to apply Theorem 2.3 to .
Finally, we prove Corollary 2.5.
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.
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Copyright © 2011 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.