- #1

RJLiberator

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## Homework Statement

Suppose X is a set with n elements. Prove that Bij(X) ≅ S_n.

## Homework 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.

## 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 don't know enough about bijections to prove this tho.