On the finite index subgroups of Houghton’s groups

Abstract

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