The Herzog-Schönheim conjecture for finitely generated groups. (English) Zbl 1468.20055
Summary: Let \(G\) be a group and \(H_1\dots,H_s\) be subgroups of \(G\) of indices \(d_1,\dots,d_s\), respectively. In 1974, Herzog and Schönheim conjectured that if \(\{H_i\alpha_i\}_{i=1}^{i=s}\), \(\alpha_i\in G\), is a coset partition of \(G\), then \(d_1,\dots,d_s\) cannot be distinct. We consider the Herzog-Schönheim conjecture for free groups of finite rank and develop a new combinatorial approach, using covering spaces. We define \(Y\) the space of coset partitions of \(F_n\) and show \(Y\) is a metric space with interesting properties. We give some sufficient conditions on the coset partition that ensure the conjecture is satisfied and moreover has a neighborhood \(U\) in \(Y\) such that all the partitions in \(U\) satisfy also the conjecture.
MSC:
| 20E05 | Free nonabelian groups |
| 20F05 | Generators, relations, and presentations of groups |
| 20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |
| 20F65 | Geometric group theory |
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