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A closure operator on the subgroup lattice of GL(๐‘›,๐‘ž) and PGL(๐‘›,๐‘ž) in relation to the zeros of the Mรถbius function

  • Luca Di Gravina EMAIL logo
From the journal Journal of Group Theory

Abstract

Let Fq be the finite field with ๐‘ž elements and consider the ๐‘›-dimensional Fq-vector space V=Fqn. In this paper, we define a closure operator on the subgroup lattice of the group G=PGLโข(V). Let ๐œ‡ denote the Mรถbius function of this lattice. The aim is to use this closure operator to characterize subgroups ๐ป of ๐บ for which ฮผโข(H,G)โ‰ 0. Moreover, we establish a polynomial bound on the number cโข(m) of closed subgroups ๐ป of index ๐‘š in ๐บ for which the lattice of ๐ป-invariant subspaces of ๐‘‰ is isomorphic to a product of chains. This bound depends only on ๐‘š and not on the choice of ๐‘› and ๐‘ž. It is achieved by considering a similar closure operator for the subgroup lattice of GLโข(V) and the same results proven for this group.

Funding statement: The author is a member of INdAM research group for Algebraic and Geometric Structures and their Applications (GNSAGA). He is also a member of the research training group GRK 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology, funded by DFG. The author thanks both groups for supporting this project.

Acknowledgements

I want to thank Francesca Dalla Volta for her careful supervision during the preparation of my Ph.D. thesis, from which this work has been extracted.

  1. Communicated by: Andrea Lucchini

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Received: 2023-02-09
Revised: 2023-08-01
Published Online: 2023-09-19
Published in Print: 2024-03-01

ยฉ 2023 Walter de Gruyter GmbH, Berlin/Boston

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