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Odd order groups with the rewriting property \(Q_3\). (English) Zbl 1011.20024
Let \(n>1\) be an integer and let \(G\) be a group. An \(n\)-subset \(\{x_1,\dots,x_n\}\) of \(G\) is said to be rewritable if there exist \(\pi\neq\sigma\) in \(S_n\) such that \(x_{\pi(1)}\cdots x_{\pi(n)}=x_{\sigma(1)}\cdots x_{\sigma(n)}\). If all \(n\)-subsets are rewritable, \(G\) has the rewriting property \(Q_n\). It was shown earlier by the author that a group has some rewriting property \(Q_n\) if and only if it is finite-by-Abelian-by-finite. Clearly \(Q_2\) is simply commutativity, while it is known that \(Q_4\)-groups are soluble. Here the author proves that a finite group \(G\) of odd order has the rewriting property \(Q_3\) if and only if \(|G'|\leq 5\).

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20F12 Commutator calculus
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