On the finite index subgroups of Houghton’s groups

Abstract Houghton’s groups $$H_2, H_3, \ldots $$ H 2 , 3 … are certain infinite permutation acting on a countably set; they have been studied, among other things, for their finiteness properties. In this note we describe all of the finite index subgroups each Houghton group, and isomorphism types. Using standard notation that d ( G ) denotes minimal size generating set , then show, $$n\in \{2, 3,\ldots \}$$ n ∈ { } U in $$H_n$$ $$d(U)\in \{d(H_n), d(H_n)+1\}$$ d ( U ) + 1 characterise when these cases occurs.