Invariable generation and the Houghton groups

Abstract

The Houghton groups H1,H2,… are a family of infinite groups. In 1977 Wiegold showed that H3 was invariably generated (IG) but H1⩽H3 was not. A natural question is then whether the groups H2,H3,… are all IG. Wiegold also ends by saying that, in the examples he had found of an IG group with a subgroup that is not IG, the subgroup was never of finite index. Another natural question is then whether there is a subgroup of finite index in H3 that is not IG. In this note we prove, for each n∈{2,3,…}, that Hn and all of its finite index subgroups are IG.The independent work of Minasyan and Goffer-Lazarovich in June 2020 frames this note quite nicely: they showed that an IG group can have a finite index subgroup that is not IG.