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A computer algebra solution to a problem in finite groups. (English) Zbl 1071.20025

The author reports on an attempt to describe a sequence \(U_1,U_2,\dots\) of words in two variables such that the finite group \(G\) is soluble iff the identity \(U_n=1\) holds in \(G\) for all but finitely many \(n\). The case of the Suzuki groups is left open. For a complete result see T. Bandman, G.-M. Greuel, F. Grunewald, B. Kunyavskij, G. Pfister and E. Plotkin [C. R., Math., Acad. Sci. Paris 337, No. 9, 581-586 (2003; Zbl 1047.20014)].
Reviewer: Rolf Brandl (Hof)

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20-04 Software, source code, etc. for problems pertaining to group theory
14G05 Rational points
20E10 Quasivarieties and varieties of groups
20F12 Commutator calculus

Citations:

Zbl 1047.20014

References:

[1] Aubry, Y. and Perret, M.: A Weil theorem for singular curves. In Arithmetic, Geometry and Coding Theory (Luminy, 1993), 1-7. De Gruyter, Berlin-New York, 1996. · Zbl 0873.11037
[2] Bandman, T., Greuel, G.-M., Grunewald, F., Kun- yavskii, B., Pfister, G. and Plotkin, E.: Engel-like identities characterising finite solvable groups. Available at http://arxiv.org/abs/math.GR/0303165.
[3] Bombieri, E.: Thompson’s problem \sigma 2 = 3. Invent. Math. 58 (1980), 77-100. · Zbl 0442.20016 · doi:10.1007/BF01402275
[4] Greuel, G.-M. and Pfister, G.: A Singular Introduction to Commutative Algebra. With contributions by O. Bchmann, C. Lossen and H. Schönemann. Springer Verlag Berlin, Heidelberg, New York, 2002. · Zbl 1023.13001
[5] Greuel, G.-M. and Pfister, G.: Computer Algebra and Finite Groups. In Mathematical software (Beijing, 2002), 4-14. World Sci. Publishing, River Edge, NJ, 2002. G.-M. Greuel · Zbl 1066.68163
[6] Greuel, G.-M., Pfister, G. and Schönemann, H.: Singular, 2-0-3. A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern, 2001. http://www.singular.uni-kl.de. · Zbl 1344.13002
[7] Plotkin, B., Plotkin, E. and Tsurkov, A.: Geometrical equiva- lence of groups. Comm. Algebra 27 (1999), 4015-4025. · Zbl 1007.20023 · doi:10.1080/00927879908826679
[8] Thompson, J.: Non-solvable finite groups all of whose local sub- groups are solvable. Bull. Amer. Math. Soc. 74 (1968), 383-437. · Zbl 0159.30804 · doi:10.1090/S0002-9904-1968-11953-6
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