On 2-local automorphisms of symmetric groups

Abstract

AbstractLet $$\phi $$ ϕ be a self-map of the symmetric group $$S_\Omega $$ S Ω acting on a countable set $$\Omega $$ Ω . We show in an elementary fashion that if $$\phi $$ ϕ has the property that $$\sigma \tau $$ σ τ is conjugate to $$\phi (\sigma )\phi (\tau )$$ ϕ ( σ ) ϕ ( τ ) for every choice of $$\sigma ,\tau $$ σ , τ in $$S_\Omega $$ S Ω , then it is necessarily an automorphism or an antiautomorphism of $$S_\Omega $$ S Ω . As a corollary to this result, the so-called 2-local automorphisms of $$S_\Omega $$ S Ω are also described.