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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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