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Semisimple groups with the rewriting property $$Q_ 5$$. (English) Zbl 0831.20027
Let $$n$$ be an integer greater than 1, and let $$G$$ be a group. A subset $$\{x_1,x_2,\dots,x_n\}$$ of $$n$$ elements of $$G$$ is said to be rewritable if there are distinct permutations $$\pi$$ and $$\sigma$$ of $$\{1,2,\dots,n\}$$ such that $$x_{\pi(1)}x_{\pi(2)}\dots x_{\pi(n)}=x_{\sigma(1)}x_{\sigma (2)}\dots x_{\sigma (n)}$$. The group $$G$$ is said to have the rewriting property $$Q_n$$ if every subset on $$n$$ elements of $$G$$ is rewritable. The authors show that the only nontrivial semisimple groups with the property $$Q_5$$ are $$A_5$$, $$S_5$$, $$\text{PSL}(2,7)$$ and $$\text{PGL}(2,7)$$. Earlier the first author [in J. Algebra 119, 246-259 (1988; Zbl 0663.20036)] showed that all groups with $$Q_4$$ are soluble.

##### MSC:
 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D06 Simple groups: alternating groups and groups of Lie type 20D05 Finite simple groups and their classification
##### Keywords:
rewritable subsets; rewriting property; semisimple groups
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##### References:
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