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Characterization of Abelian-by-cyclic \(3\)-rewritable groups. (English) Zbl 1138.20035
Summary: Let \(n\) be an integer greater than 1. A group \(G\) is said to be \(n\)-rewritable (or a \(Q_n\)-group) if for every \(n\) elements \(x_1,\dots,x_n\) in \(G\) there exist distinct permutations \(\sigma\) and \(\tau\) in \(S_n\) such that \(x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(n)}=x_{\tau(1)}x_{\tau(2)}\cdots x_{\tau(n)}\). In this paper, we completely characterize Abelian-by-cyclic 3-rewritable groups: they turn out to have an Abelian subgroup of index 2 or the size of derived subgroups is less than 6. In this paper, we also prove that \(G/F(G)\) is an Abelian group of finite exponent dividing 12, where \(F(G)\) is the Fitting subgroup of \(G\).

MSC:
20F05 Generators, relations, and presentations of groups
20E07 Subgroup theorems; subgroup growth
20F16 Solvable groups, supersolvable groups
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