Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-09T07:34:55.308Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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