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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.

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
Full Text: DOI
[1] Aschbacher M., Finite group theory (1986) · Zbl 0583.20001
[2] DOI: 10.1016/0021-8693(88)90233-5 · Zbl 0647.20033
[3] 1988.rewriting products of group elements-II, 246–259. J.Algebra 119.
[4] Blyth, R. D. and Robinson, D. J. S. Recent progress on rewritability in groups. Group Theory, Proceedings of the 1987 Singapore conference. de Gruyter. Berlin · Zbl 0663.20037
[5] 1990.Solution of the solubility problem for rewritable groups, 438–444. J.London Math.Soc.(2)41.
[6] Pure, J., ed. 1991.Insoluble groups with the rewriting property P8, 251–263. Appl.Algebra 72.
[7] Brandl R., Arch. Math. (Basel) 53 pp 245– (1989)
[8] Carter R. W., Simple groups of Lie type (1972) · Zbl 0248.20015
[9] Conway J. H., ATLAS of Finite Groups (1985)
[10] Curzio M., Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur 53 pp 136– (1983)
[11] DOI: 10.1007/BF01229319 · Zbl 0544.20036
[12] Huppert B., Endliche Gruppen I (1967) · Zbl 0217.07201
[13] Huppert B., Finite Groups III (1982) · Zbl 0514.20002
[14] Longobardi P., The classification of groups in which every product of four elements can be reordered · Zbl 0838.20038
[15] Longobardi P., Illinois J. Math 35 pp 198– (1991)
[16] DOI: 10.1016/0021-8693(91)90066-H · Zbl 0721.20022
[17] DOI: 10.4153/CJM-1990-056-4 · Zbl 0727.20027
[18] DOI: 10.1090/S0002-9947-1982-0654839-1
[19] DOI: 10.2307/1970423 · Zbl 0106.24702
[20] Tits J., Sém. Bourbaki 13 pp 210– (1960)
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