SOME REMARKS ON THE DIMENSION OF SUBGROUPS IN A DESCENDING AND ASCENDING SERIES OF SOLVABLE ALGEBRAIC GROUPS

Ngô Thị Ngoan; Nguyễn Quốc Linh; Nguyễn Quốc Thắng; Phạm Đức Hiệp

doi:10.34238/tnu-jst.12905

SOME REMARKS ON THE DIMENSION OF SUBGROUPS IN A DESCENDING AND ASCENDING SERIES OF SOLVABLE ALGEBRAIC GROUPS

About this article

Received: 28/05/25 Revised: 21/08/25 Published: 21/08/25

Authors

1. Ngo Thi Ngoan , Thuongmai University
2. Nguyen Quoc Linh, VNU University of Science
3. Nguyen Quoc Thang, Institute of Mathematics - VAST
4. Pham Duc Hiep, VNU University of Education

Abstract

In the case the linear algebraic groups under consideration are connected nilpotent or solvable algebraic groups, there were some well known estimations on dimension of subgroups in the descending derived series and ascending central series for connected solvable or nilpotent algebraic groups, which are very useful when one uses mathematical induction to investigate the structure of the given linear algebraic groups. Our aim in this note is to investigate to what extent one can extend these estimates to the case of non- connected linear algebraic solvable or nilpotent groups. Our methods use, besides the standard results from the theory of linear algebraic groups, some generalizations of a Schur’s lemma and a Baer’s lemma to non-connected linear algebraic groups. Our results are some generalizations of these estimates to the case of not necessarily connected nilpotent or solvable algebraic groups. Besides, we also give some applications and examples, in order to show that some of our results are optimal. The main results of the paper bring out interesting characteristics of solvable or nilpotent algebraic groups defined over an algebraically closed field.

Keywords

Nilpotent groups; Solvable groups; Dimension; Ascending series; Descending series

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References

[1] A. Borel, Linear Algebraic Groups, Second enlarged ed., Graduate Texts in Math., vol. 126, Springer-Verlag, New York, 1991.

[2] M. Demazure and P. Gabriel, Groupes algébriques, Tome I, Paris, Masson, 1970.

[3] J. E. Humphreys, Linear Algebraic Groups, Second ed., Graduate Texts in Math., vol. 51, Springer- Verlag, New York, 1981.

[4] J. S. Milne, Algebraic Groups. The theory of group schemes of finite type over a field, Cam- bridge Studies in Advanced Mathematics, vol. 170, Cambridge University Press, 2017.

[5] S. Gelaki, "Twisting of Affine Algebraic Groups. I," International Mathematics Research Notices, no. 16, pp. 7552–7574, 2015.

[6] S. Gelaki, "Twisting of Affine Algebraic Groups. II," International Mathematics Research Notices, no. 11, pp. 8508–8539, 2022.

[7] Y. Peterzil and S. Starchenko, "O-minimal flows on nilmanifolds," Duke Mathematical Journal, vol. 170, no. 18, pp. 3935–3976, 2021.

[8] J. P. Labesse, "Cohomologie, stabilisation et changement de base,” Astérisque, no. 257, pp. 1-116, 1999.

[9] P. N. Achar, W. Hardesty, and S. Riche, "Representation Theory of Disconnected Reductive Groups," Documenta Mathematica, vol. 25, pp. 2149–2177, 2020.

[10] J. J. Rotman, An Introduction to the Theory of Groups, 5th ed., Graduate Texts in Mathematics, vol. 148, Springer, 1999.

[11] D. T. Nguyen and Q. T. Nguyen, "An analog of Serre’s conjectures, Galois cohomology and defining equation of unipotent groups," Proc. Jap. Acad., vol. 83, no. 7, ser. A, pp. 93-98, 2007.

[12] D. T. Nguyen and Q. T. Nguyen, "Galois cohomology of unipotent groups and field extensions," Commun. in Algebra, vol. 39, pp. 1-16, 2011.

[13] V. P. Platonov, "On the theory of algebraic groups," Doklady Acad. Sci., vol. 146, pp. 1025-1026, 1962.

DOI: https://doi.org/10.34238/tnu-jst.12905

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