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Rewriting products of group elements. I. (English) Zbl 0647.20033
A group G has Property \(Q_ n\) (n\(\geq 2)\) if for every choice of elements \(x_ 1,...,x_ n\) there exist distinct permutations \(\sigma\) and \(\tau\) on n letters such that \[ x_{\sigma (1)}...x_{\sigma (n)}=x_{\tau (1)}...x_{\tau (n)}. \] If \(\sigma\) can always be chosen to be the identity, then G is said to have property \(P_ n\) [see M. Curzio, P. Langobardi, M. Maj and D. J. S. Robinson, Arch. Math. 44, 385-389 (1985; Zbl 0544.20036)].
The following is proved: Theorem. A group has \(Q_ n\) for some n if and only if it is finite-by-abelian-by-finite. Clearly, \(P_ 2=Q_ 2=commutativity\) and \(P_ n\subseteq Q_ n\), the last inclusion being strict for \(n\geq 3\) (Proposition 2.10). By the analogue to the Theorem for \(P_ n\) (loc. cit.), every group G in \(Q_ n\) has \(P_ m\) for some \(m=f(n,G)\). The analogous statement is false for semigroups [see G. Pirillo, in: Group Theory, Proc. Conf., Brixen/Italy 1986, Lect. Notes Math. 1281, 118-119 (1987; Zbl 0629.20028)]. It seems to be an open question whether \(Q_ n\subseteq P_{f(n)}\) for some function f (compare with Proposition 2.11).
Reviewer: R.Brandl

20F12 Commutator calculus
20F24 FC-groups and their generalizations
20E10 Quasivarieties and varieties of groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20F05 Generators, relations, and presentations of groups
Full Text: DOI
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