The Houghton groups \(H_{1},H_{2}, \ldots\) are a family of infinite groups: the group \(H_{1}\) is isomorphic to \(\mathrm{FSym}(\mathbb{Z})\), the finitary symmetric group over \(\mathbb{Z}\), and \(H_{2}\) is a semidirect product of \(H_{1}\) and the infinite cyclic group. For a precise definition of \(H_{n}\) for \(n \geq 3\), the reader can see the paper under review. It is important to remark that \(H_{n} \leq \mathrm{FSym}(X_{n})\), where \(X_{n}=\{1,\ldots,n\} \times \mathbb{N}\). Furthermore there is a short exact sequence of groups \[1 \longrightarrow \mathrm{FSym}(X_{n}) \longrightarrow H_{n} \overset{\pi}{\longrightarrow} \mathbb{Z}^{n-1} \longrightarrow 1\] where \(\pi\) is induced by defining \(\pi(g_{i}):= e_{i-1}\) for \(i \in \{2,\ldots , n\}\), where \(e_{i}\) denotes the vector in \(\mathbb{Z}^{n-1}\) with \(i\)th entry \(1\) and other entries \(0\).
In this note, the author describes all of the finite index subgroups of each Houghton group and their isomorphism types (Theorem 1).
If \(G\) is a finitely generated group let \(d(G)=\min \{ |S| \mid S\subset G, \langle S \rangle =G \}\). To denote that \(H\) is a subgroup of \(G\) and \(|G:H| < \infty\), in this paper it is written \(H \leq_{\mathsf{f}} G\). The following interesting result is proved
Theorem 3: If \(U \leq_{\mathsf{f}} H_{2}\), then \(d(U) = d(H_{2})\). For \(n\in \{3, 4, \ldots \}\) and \(U \leq_{\mathsf{f}} H_{n}\), we have that \(d(U)\in \{d(H_{n}), d(H_{n}) + 1\}\). Furthermore, let \(\pi(U) =\langle c_{2}e_{1},\ldots , c_{n}e_{n-1}\rangle\). Then \(d(U) = d(H_{n}) + 1\) occurs exactly when both of the following conditions are met: (i) that \(\mathrm{FSym}(X_{n}) \leq U\); and (ii) either one or zero elements in \(\{c_{2}, \ldots , c_{n}\}\) are odd.