# zbMATH — the first resource for mathematics

The width of a power of a free nilpotent group of nilpotency class $$2$$. (English. Russian original) Zbl 0956.20028
Sib. Math. J. 41, No. 1, 173-179 (2000); translation from Sib. Mat. Zh. 41, No. 1, 206-213 (2000).
Let $$G$$ be a group. Let $$wG$$ be the verbal subgroup of $$G$$ defined by a word $$w$$. Recall that the width $$\text{wid}(g,w)$$ of $$g\in wG$$ with respect to $$w$$ is the least number $$l$$ such that $$g$$ is equal to a product of $$l$$ values of $$w^{\pm 1}$$ in $$G$$. For a subset $$M$$ of $$wG$$, define $$\text{wid}(g,M)$$ by $$\max_{g\in M}\text{wid}(g,w)$$.
The author proves that the width of a free nilpotent finitely generated group of class 2 with respect to $$x^k$$ is finite and does not depend on $$k$$. More precisely, the following theorem is proven: Let $$N_n$$ be a free nilpotent group of class 2 and let $$k\geq 1$$ be a positive integer. Then (1) $$\text{wid}(x^{2k},N_n^{2k})=2[n/2]+1$$, where $$n\geq 2$$; (2) $$\text{wid}(x^{2k+1},N_n^{2k+1})=1$$.

##### MSC:
 20F18 Nilpotent groups 20F05 Generators, relations, and presentations of groups 20F12 Commutator calculus 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
Full Text:
##### References:
 [1] Merzlyakov Yu. I., ”Algebraic linear groups as divisible groups of automorphisms and closure of their verbal subgroups,” Algebra i Logika,6, No. 1, 83–94 (1967). [2] Merzlyakov Yu. I., Rational Groups [in Russian], Nauka, Moscow (1987). · Zbl 0647.20042 [3] Malcev A. I., ”On free solvable groups,” Dokl. Akad. Nauk SSSR,130, No. 3, 495–498 (1960). [4] Allambergenov Kh. S. and Roman’kov V. A., ”On products of commutators in groups,” submitted to VINITI, No. 4566-85. [5] Allambergenov Kh. S. andRoman’kov V. A., ”On products of commutators in groups,” Dokl. Akad. Nauk UzSSR,4, 14–15 (1981). · Zbl 0578.20025 [6] Bavard C. andMeighiez G., ”Commutateurs dans les groupes metabeliens,” Indag. Math. New Ser.,3, No. 2, 129–135 (1992). · Zbl 0769.20015 [7] Stroud P., Thesis, Cambridge Univ., Cambridge (1966). [8] Robinson D., A Course in the Theory of Groups, Springer-Verlag, Berlin, Heidelberg, and New York (1982). · Zbl 0483.20001 [9] Rhemtulla A. H., ”Commutators of certain finitely generated soluble groups,” Canad. J. Math.,21, 1160–1164 (1969). · Zbl 0186.03903 [10] Roman’kov V. A., ”On the width of verbal subgroups in solvable groups,” Algebra i Logika,21, No. 1, 60–72 (1982). · Zbl 0513.20022 [11] Wilson J., ”On outer-commutator words,” Canad. J. Math.,26, No. 3, 608–620 (1974). · Zbl 0277.20051 [12] Rhemtulla A. H., ”A problem of bounded expressibility in free products,” Proc. Cambridge Phil. Soc.,64, No. 3, 573–584 (1968). · Zbl 0159.03001 [13] Bardakov V. G., The Width of Verbal Subgroups of SomeHNN-Extensions [in Russian], [Preprint, No. 9], Inst. Mat., Rossiîsk. Akas. Nauk, Sibirsk. Otdel., Novosibirsk (1995). · Zbl 0943.20034 [14] Akhavan M. andRhemtulla A. H., ”Commutator length of abelian-by-nilpotent groups,” Glasgow Math. J.,40, No. 1, 117–121 (1998). · Zbl 0911.20028 [15] Allambergenov Kh. S., ”On the width of the commutator subgroup of a free metabelian group,” in: Abstracts: 10 All-Union Symposium on Group Theory, Minsk, 1986, p. 5. [16] Bardakov V. G., ”To the theory of braid groups,” Mat. Sb.,183, No. 6, 3–42 (1992). · Zbl 0798.20029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.