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M\'obius function of the subgroup lattice of a finite group and Euler Characteristic
- Publication Year :
- 2022
-
Abstract
- The M\"obius function of the subgroup lattice of a finite group has been introduced by Hall and applied to investigate several questions. In this paper, we consider the M\"obius function defined on an order ideal related to the lattice of the subgroups of an irreducible subgroup $G$ of the general linear group $\mathrm{GL}(n,q)$ acting on the $n$-dimensional vector space $V=\mathbb{F}_q^n$, where $\mathbb{F}_q$ is the finite field with $q$ elements. We find a relation between this function and the Euler characteristic of two simplicial complexes $\Delta_1$ and $\Delta_2$, the former raising from the lattice of the subspaces of $V$, the latter from the subgroup lattice of $G$.<br />Comment: 10 pages. Revised version; the first version contained some inaccuracies that have been corrected
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2212.01917
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s10801-024-01329-8