# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Aschbacher M., Finite group theory (1986) · Zbl 0583.20001  DOI: 10.1016/0021-8693(88)90233-5 · Zbl 0647.20033  1988.rewriting products of group elements-II, 246–259. J.Algebra 119.  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  1990.Solution of the solubility problem for rewritable groups, 438–444. J.London Math.Soc.(2)41.  Pure, J., ed. 1991.Insoluble groups with the rewriting property P8, 251–263. Appl.Algebra 72.  Brandl R., Arch. Math. (Basel) 53 pp 245– (1989)  Carter R. W., Simple groups of Lie type (1972) · Zbl 0248.20015  Conway J. H., ATLAS of Finite Groups (1985)  Curzio M., Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur 53 pp 136– (1983)  DOI: 10.1007/BF01229319 · Zbl 0544.20036  Huppert B., Endliche Gruppen I (1967) · Zbl 0217.07201  Huppert B., Finite Groups III (1982) · Zbl 0514.20002  Longobardi P., The classification of groups in which every product of four elements can be reordered · Zbl 0838.20038  Longobardi P., Illinois J. Math 35 pp 198– (1991)  DOI: 10.1016/0021-8693(91)90066-H · Zbl 0721.20022  DOI: 10.4153/CJM-1990-056-4 · Zbl 0727.20027  DOI: 10.1090/S0002-9947-1982-0654839-1  DOI: 10.2307/1970423 · Zbl 0106.24702  Tits J., Sém. Bourbaki 13 pp 210– (1960)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.