Automorphisms of Mackey groups

Authors

DOI:

https://doi.org/10.17721/1812-5409.2023/2.2

Keywords:

Automorphisms, anti-automorphisms, infinite simple finatary torsion group, Mackey group

Abstract

We consider total subspaces of linear functionals on an infinite-dimensional vector space and the related Mackey algebras and groups. We outline the description of automorphisms of Mackey groups SL(V|W), O(f), and SU(f) over fields of characteristics not equal to 2, 3. Moreover, the paper explores the relationship between field automorphisms and automorphisms of the aforementioned groups.

J.Hall proved that infinite simple finitary torsion groups are the alternating groups on infinite sets or Mackey groups over a field, which is an algebraic extension of a finite field. J.Schreier and S.Ulam described automorphisms of infinite alternating groups. With the description of automorphisms of finitary Mackey groups and special finitary unitary Mackey groups we finish classification of automorphisms of all infinite simple finitary torsion groups over fields of characteristics not equal to 2, 3. The proof is based of description of automorphisms of elementary linear groups over associative rings that due to I.Golubchik, A.Mikhalev and E.Zelmanov.

Pages of the article in the issue: 16 - 19

Language of the article: English

References

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BEZUSHCHAK, O., OLIYNYK, B. (2020) "Primary decompositions of unital locally matrix algebras", Bull. Math. Sci. 10(1). doi:10.1142/S166436072050006X

HALL, J.I. (2006) "Periodic simple groups of finitary linear transformations", Annals of Math. 163, pp.445-498. doi:10.4007/annals.2006.163.445

JACOBSON, N. (1975) "Lectures in abstract algebra. Volume 2. Linear algebra", Springer-Verlag.

MACKEY, G. (1945) "On infinite-dimensional linear spaces", Trans. AMS, 57, pp.155-207.

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SCHRERIER, J. "Uber die Permutationsgruppe der naturlichen Zahlenfolge", Studia Math., 4, pp.134-141.

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Published

2023-12-23

How to Cite

Bezushchak, O. (2023). Automorphisms of Mackey groups. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, (2), 16–19. https://doi.org/10.17721/1812-5409.2023/2.2

Issue

Section

Algebra, Geometry and Probability Theory