Some embeddings between symmetric R. thompson groups

Let $m\leqslant n\in \mathbb {N}$ , and $G\leqslant \operatorname {Sym}(m)$ and $H\leqslant \operatorname {Sym}(n)$ . In this article, we find conditions enabling embeddings between the symmetric R. Thompson groups ${V_m(G)}$ and ${V_n(H)}$ . When $n\equiv 1 \mod (m-1)$ , and under some other technical conditions, we find an embedding of ${V_n(H)}$ into ${V_m(G)}$ via topological conjugation. With the same modular condition, we also generalize a purely algebraic construction of Birget from 2019 to find a group $H\leqslant \operatorname {Sym}(n)$ and an embedding of ${V_m(G)}$ into ${V_n(H)}$ .