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Schur-Zassenhaus theorem revisited

Published online by Cambridge University Press:  12 March 2014

Alexandre V. Borovik  and
Ali Nesin
Alexandre V. Borovik
Affiliation:
Department of Mathematics, University of Manchester Institute of Science and Technology, Manchester M60 1QD, United Kingdom
Ali Nesin
Affiliation:
Department of Mathematics, University of California, Irvine California 92717, E-mail: anesin@math.uci.edu
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Extract

One of the purposes of this paper is to prove a partial Schur-Zassenhaus Theorem for groups of finite Morley rank.

Theorem 2. Let G be a solvable group of finite Morley rank. Let π be a set of primes, and let H ⊲ G a normal π-Hall subgroup. Then H has a complement in G.

This result has been proved in [1] with the additional assumption that G is connected, and thought to be generalized in [2] by the authors of the present article. Unfortunately in the last section of the latter paper there is an irrepairable mistake. Here we give a new proof of the Schur-Zassenhaus Theorem using the results of [2] up to the last section and a new result that we are going to state below.

The second author has shown in [11] that a nilpotent ω-stable group is the central product of a divisible subgroup and a subgroup of bounded exponent, generalizing a well-known result of Angus Macintyre about abelian groups [8]. One could ask a similar question for solvable groups: are they a product of two subgroups, one divisible, one of bounded exponent? One is allowed to be hopeful because of the well-known decomposition of the connected solvable algebraic groups over algebraically closed fields as the product of the unipotent radical and a torus.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 1 , March 1994 , pp. 283 - 291
Copyright
Copyright © Association for Symbolic Logic 1994

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References

REFERENCES

[1] Altinel, T., Cherlin, G., Corredor, L.-J., and Nesin, A., A Hall theorem for ω-stable groups, submitted.Google Scholar
[2] Borovik, A. V. and Nesin, A., On the Schur-Zassenhaus theorem for groups of finite Morley rank, this Journal, vol. 57 (1992), pp. 1468–1477.Google Scholar
[3] Borovik, A. V. and Nesin, A., Groups of finite Morley rank, Oxford University Press (to appear).CrossRefGoogle Scholar
[4] Borovik, A. V. and Poizat, B. P., Tores et p-groupes, this Journal, vol. 55 (1990), pp. 478–491.Google Scholar
[5] Brown, K., Cohomology of groups, Springer-Verlag, New York, 1982.CrossRefGoogle Scholar
[6] DeBonis, M. J. and Nesin, A., On split Zassenhaus groups of mixed characteristic and of finite Morley rank, Journal of the London Mathematical Society (to appear).Google Scholar
[7] Kegal, O. and Wehrfritz, B., Locally finite groups, North-Holland, Amsterdam, 1973.Google Scholar
[8] Macintyre, A., On ω 1-categorical theories of abelian groups, Fundamenta Mathematicae, vol. 70 (1971), pp. 253–270.CrossRefGoogle Scholar
[9] Nesin, A., Solvable groups of finite Morley rank, Journal of Algebra, vol. 121 (1989), pp. 26–39.CrossRefGoogle Scholar
[10] Nesin, A., On solvable groups of finite Morley rank, Transactions of the American Mathematical Society, vol. 321 (1990), pp. 659–690.CrossRefGoogle Scholar
[11] Nesin, A., Poly-separated and ω-stable nilpotent groups, this Journal, vol. 56 (1991), pp. 694–699.Google Scholar
[12] Poizat, B., Groupes stables, Nur al-Mantiq wal—Ma'rifah, Villeurbanne, France, 1987, (French)Google Scholar
[13] Suzuki, M., Group theory. I, vol. 247, Springer-Verlag, 1982.CrossRefGoogle Scholar
[14] Zil'ber, B. I., Groups with categorical theories, Abstracts of papers presented at the fourth all-union symposium on group theory, Matematicheskogo Instituta Sibirsk, Otdeleniya Akademii Nauk SSSR, 1973, pp. 63–68, (Russian)Google Scholar
[15] Zil'ber, B. I., Groups and rings whose theory is categorical, Fundamenta Mathematicae, vol. 55 (1977), pp. 173–188.Google Scholar