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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

##### MSC:
 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
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##### References:
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