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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) \(\text{wid}(x^{2k+1},N_n^{2k+1})=1\).

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: DOI EuDML
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