Let \(G\) be a group and \(\varphi \in \operatorname{Aut}(G)\). With the understanding that all actions are right actions, two elements \(a\) and \(b\) of \(G\) are \(\varphi\)-twisted conjugate, denoted \(a \sim_{\varphi} b\) is there is \(x \in G\) such that \(b = (x^{-1} \varphi) a x.\) The Reidemeister number \(R(\varphi)\) is the number of \(\varphi\)-twisted conjugacy classes and \(G\) is set to be an \(R_{\infty}\)-group if \(R(\varphi)\) is infinite whenever \(\varphi \in \operatorname{Aut}(G)\). Examples of \(R_{\infty}\)-groups are Houghton’s groups [C. H. Houghton, Arch. Math. 31, 254–258 (1978; Zbl 0377.20044); J. H. Jo et al., Algebra Discrete Math. 23, No. 2, 249–262 (2017; Zbl 1381.20032)]. Given a set \(X\), the group of permutations of \(X\) with finite support and its subgroup of even permutations are denoted by \(\mathrm{FSym}(X)\) and \(\mathrm{FAlt}(X)\), respectively. The paper has a clutch of results (Corollaries \(3.4\)–\(3.7\)) showing that a subgroup \(G\) of \(\mathrm{Sym}(X)\) fully containing \(\mathrm{FAlt}(X)\) has the \(R_{\infty}\)-property assuming varying additional conditions. The key to the proofs is contained in Lemma \(2.4\) and Proposition \(2.5\), stating that \(\mathrm{FAlt}(X) \, \mathrm{char} \, G\) and that therefore \(\operatorname{Aut}(G) \cong N_{\mathrm{Sym}(X)}(G)\) (compare Corollary 3.3 in [D. Gonçalves and P. Sankaran, Pac. J. Math. 280, No. 2, 349–369 (2016; Zbl 1383.20026)]). Lemma \(3.8\), stating that if \(G \leq \mathrm{Sym}(X)\) is torsion, then so is \(\langle G,\, \mathrm{FAlt}(X) \rangle\) opens a door to further results, stating that there are ”many” \(R_{\infty}\)-torsion groups. Corollary \(3.10\) says that, given a torsion group \(G\) and a cardinal \(\alpha \geq \vert G \vert\) there is an \(R_{\infty}\)-torsion group of cardinality \(\alpha\) containing an isomorphic copy of \(G\); uncountable families of countable and finitely generated \(R_{\infty}\)-torsion groups are constructed in Corollaries \(3.11\) and \(3.12\), respectively.
Two groups \(G\) and \(H\) are commensurable if they have isomorphic finite-index normal subgroups with isomorphic quotients. The last section of the paper establishes Theorem \(4.6\): If \(G\) is a group commensurable to \(H_n\), the \(n\)th Houghton group (\(n \geq 2)\), then \(G\) has the \(R_{\infty}\)- property.