Double coset decomposition of groups
The author shows that if \(G\) is a group which is either residually finite or word hyperbolic and if \(G\) can be decomposed as a finite union of double cosets of a cyclic subgroup, then \(G\) is virtually cyclic. The work answers for a large class of groups a question of M. Dunwoody. The author applies then this result to the study of Frobenius groups (that is, groups acting transitively on a set of cardinality \(\geq 2\) such that the only element which fixes two distinct points is the identity). He shows finally that if \(G\) is an arbitrary group and if \(H\) is a subgroup of \(G\) and if \(G\) is a finite union of double cosets of \(H\), then no subgroup of \(G\) can split over a subgroup commensurable with \(H\).
scientific article; zbMATH DE number 1384472