On two Möbius functions for a finite non-solvable group

Abstract

Let G be a finite group, μ be the Möbius function on the subgroup lattice of G, and λ be the Möbius function on the poset of conjugacy classes of subgroups of G. It was proved by Pahlings that, whenever G is solvable, the property $\mu (H,G)=[N_{G\prime }(H):G^{\prime }\cap H]·\lambda (H,G)$ holds for any subgroup H of G. It is known that this property does not hold in general, the Mathieu group M₁₂ being a counterexample. In this paper we investigate the relation between μ and λ for some classes of non-solvable groups, among them, the minimal non-solvable groups. We also provide several examples of groups not satisfying the property.

Keywords:

• Möbius function
• non-solvable group
• subgroup lattice

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D30
• 05E15
• 06A07

Acknowledgements

We thank Emilio Pierro for some useful remarks about the Ree groups.

Notes

1 Pay attention to a misprint in the proof of [18, Proposition 3]: the right value of a_n is 1 or 2 according respectively to $p\equiv \pm 1$ or $p\cancel{\equiv }\pm 1$ modulo 4n, not modulo 2n as it is written.

Additional information

Funding

This research was partially supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GSAGA - INdAM). The second author is funded by the project “Attrazione e Mobilità dei Ricercatori” Italian PON Programme (PON-AIM 2018 num. AIM1878214-2). Ministero dell'Università e della Ricerca.