## TYI 25: The Automorphism Group of the Symmetric Group

True or False: The group of automorphisms of the symmetric group $S_n$, n ≥ 3 is $S_n$ itself.

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### 7 Responses to TYI 25: The Automorphism Group of the Symmetric Group

1. James says:

The answer is hard to believe. It reminds me of the smooth structures on \$\mathbb{R}^n\$.

2. ameyerowitz says:

To have anything else happen would be as unlikely as a perfect number which is also a factorial!

3. aram says:

I guess you could say that my intuition was correct almost everywhere. 🙂

4. Gil Kalai says:

Dear James, Aaron, and Aram – thanks for the comments!

5. Paco Santos says:

Coincidentally (?), two days after this was posted John McCammond uploaded a nice little paper to the arXiv containing the answer.