21.83 (2026)

The function $d_n(\sigma, \tau) = (1/n) \cdot |\{x \in \{1, \dots, n\} \mid \sigma(x) \neq \tau(x)\}|$ is a distance on the symmetric group $S_n$. For a finitely generated group $G$, an almost-homomorphism is a sequence of set-theoretic maps $f_n : G \to S_n$ satisfying $d_n(f_n(g)f_n(h), f_n(gh)) \to 0$ as $n \to \infty$ for all $g, h \in G$. An almost-homomorphism $\{f_n\}$ is said to be close to a homomorphism if there is a sequence of group homomorphisms $\rho_n : G \to S_n$ such that $d_n(\rho_n(g), f_n(g)) \to 0$ as $n \to \infty$ for all $g \in G$. The group $G$ is said to be permutation stable if every almost-homomorphism of $G$ is close to a homework.

Conjecture: Metabelian groups are permutation-stable.