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Group theory via global semigroup theory. (English) Zbl 0668.20048

Groups are shown to be special homomorphic images of residually finite inverse semigroups. To this aim the authors define three types of expansions of semigroups. When the starting semigroup is a group the three types of expansions give isomorphic semigroups and (selecting a large enough generator set for the group) the expansion thus obtained is an inverse semigroup.
The results are applied to the Burnside problem and the following proposition is obtained: Let G be a finite group of odd order. Then G is a homomorphic image of a bounded torsion semigroup the finite subgroups of which are cyclic.
Reviewer: T.J.Harju

MSC:

20M10 General structure theory for semigroups
20M05 Free semigroups, generators and relations, word problems
20F05 Generators, relations, and presentations of groups
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[1] Birget, J. C.; Rhodes, J., Almost finite expansions of arbitrary semigroups, J. Pure Appl. Algebra, 32, 249-287 (1984) · Zbl 0546.20055
[2] Clifford, A. H.; Preston, G. B., The algebraic theory of semigroups, (Mathematical Surveys 7, Vol. 2 (1967), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 1967 · Zbl 0178.01203
[3] Rhodes, J., Infinite iteration of matrix semigroups, J. Algebra, 98, 422-451 (1986), Part I · Zbl 0584.20053
[4] Margolis, S.; Pin, J. E., Expansions, free inverse semigroups, and Schützenberger product, J. Algebra, 110, 2, 298-305 (1987) · Zbl 0625.20044
[5] Munn, W. D., Free inverse semigroups, (Proc. London Math. Soc., 29 (1974)), 385-404, (3) · Zbl 0305.20033
[6] Scheiblich, H. E., Free inverse semigroups, (Proc. Amer. Math. Soc., 38 (1973)), 1-7 · Zbl 0256.20079
[7] Schein, B., Free inverse semigroups are not finitely presentable, Acta Math. Sci. Hungar., 26, 41-52 (1975) · Zbl 0301.20041
[8] Petrich, M., Inverse Semigroups (1984), Wiley-Interscience: Wiley-Interscience New York · Zbl 0546.20053
[9] Adian, S. I., The Burnside problem and identities in groups, (Ergeb. Math. Grenzgeb., Vol. 95 (1980), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 1343.20040
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