The space of coset partitions of F_n and Herzog-Schönheim conjecture

2018/04/30 by Chouraqui, Fabienne
#Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1804.11103

Abstract

Let G be a group and H₁,...,H_s be subgroups of G of indices d₁,...,d_s respectively. In 1974, M. Herzog and J. Schönheim conjectured that if \H_iα_i\_i=1^i=s, α_i∈ G, is a coset partition of G, then d₁,..,d_s cannot be distinct. We consider the Herzog-Schönheim conjecture for free groups of finite rank. We define Y the space of coset partitions of F_n and show Y is a metric space with interesting properties. In a previous paper, we gave some sufficient conditions on the coset partition of F_n that ensure the conjecture is satisfied. Here, we show that each coset partition of F_n, which satisfies one of these conditions, has a neighborhood U in Y such that all the partitions in U satisfy also the conjecture.

Related

• The Herzog-Schönheim conjecture for solvable groups
• The Herzog-Schönheim conjecture for simple and symmetric groups
• On the Herzog-Schönheim conjecture for uniform covers of groups
• The Herzog-Sch "onheim Conjecture for small groups and harmonic\n subgroups
• Decomposition of locally compact coset spaces