# zbMATH — the first resource for mathematics

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 $$|G'|\leq 5$$.

##### MSC:
 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20F12 Commutator calculus
##### Keywords:
rewritable groups; permutation properties; odd order groups
Full Text: