1. The problem statement, all variables and given/known data Suppose X is a set with n elements. Prove that Bij(X) ≅ S_n. 2. Relevant equations S_n = Symmetric set ≅ = isomorphism Definition: Let G and G2 be groups. G and G2 are called Isomorphic if there exists a bijection ϑ:G->G2 such that for all x,y∈G, ϑ(xy) = ϑ(x)ϑ(y) where the LHS is operation in G and the RHS is operation in G2. 3. The attempt at a solution So if we have a set X with n elements. A Bijection simply sends one element to some other unique element. The symmetric operation just sends one element to a unique other element as well. So clearly both sides have unique elements. IF we take ϑ(xy) in the Bij(x) that sends them to ϑ(x)ϑ(y) in the symmetric group I dont know enough about bijections to prove this tho.