Is there exist a group homomorphism from the symmetric group $S_n$ to $S_{n-1}$ for $n \ge 5?$

Question

Does there exist a group homomorphism from the symmetric group $S_n$ to $S_{n-1}$ for $n \ge 5?$

My attempt: I think not, because for $n \ge 5$ , $A_n$ is the only normal subgroup of $S_n$.

Is it true?

Answer 1

As you are aware that $A_n$ is a normal subgroup how about trying for a homomorphism that will have $A_n$ as its kernel.

Take any permutation $\tau\in S_{n-1}$ of order 2. For example $\tau =(17)(38)(26)\in S_8$. (cycle notation).

Define for $\sigma\in A_n$, $\phi(\sigma) =id$, and $\phi(\sigma)=\tau$ for $\sigma\in S_n-A_n$. This is a nontrivial homomorphism with image a subgroup of order $2$. This shows that in fact there are as many homomorphims as there are elemnts of order 2 in $S_{n-1}$. The same argument works by replacing the codomain $S_{n-1}$ by any group of even order (they always have elements of order 2).