Prove that there exists at least one element $f$ in $S_4$ such that $f$ cannot be expressed as $g^4$ for any $g\in S_4$.

Question

Given that $S_4$ is the set of all permutations on $\{1,2,3,4\}$. Prove that there exists at least one element $f$ in $S_4$ such that $f$ cannot be expressed as $g^4$ for any $g\in S_4$.

So if I construct a function $\mathcal{F}:S_4\to S_4$ and show that this is not surjective, then the problem will be solved. But I cannot construct such a function. Any assistance would be helpful.

Addition: Since $S_4$ is finite, we can also prove that $\mathcal{F}$ is not injective which will imply that $\mathcal{F}$ is not surjective.

Answer 1

Define the function $$ F:S_4 \to S_4$$ as $$ F(f)=f^4$$ The function F is not injective because for example for $$f=(1,2,3,4)$$ we have $$F(f)=id$$ and $$F(id)=id$$ where $id$ is the identity function $$id(x)=x$$ Thus $F$ is not surjective as well and we are done.

Answer 2

Consider the types of factors your permutations can have: You can have identity, you can have a transposition $(a,b)$, you can have triplets $(a,b,c)$ or a last $(a,b,c,d)$.

Now, clearly $(a,b)^4 =1$, $(a,b,c)^4 = (a,b,c)^3(a,b,c) = 1(a,b,c)$ and $(a,b,c,d)^4 = 1$.

So $S_4^4$ consists solely of permutations of the form $(a,b,c)$ and the identity.