The Möbius function of the small Ree groups

2014/10/31 by Pierro, Emilio
#11A25 #20B20 #20E15 #20G41 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1410.8702

Abstract

The Möbius function for a group, G, was introduced in 1936 by Hall in order to count ordered generating sets of G. In this paper we determine the Möbius function of the simple small Ree groups, R(q)=²G₂(q) where q=3^2m+1 for m>0, using their 2-transitive permutation representation of degree q³+1 and describe their maximal subgroups in terms of this representation. We then use this to determine \vertEpi(Γ,G)\vert for various Γ, such as F₂ or the modular group PSL₂(ℤ), with applications to Grothendieck's theory of dessins d'enfants as well as probabilistic generation of the small Ree groups.

Related

• The Möbius function of \rm PSU(3,2²ⁿ)
• The Möbius function of \rm PSL(3,2^p) for any prime p
• On quantitative structure of small Ree groups
• The Möbius Function of the Suzuki Groups
• The Möbius Number of the Socle of any Group