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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
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References:
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