Show that ${\rm Inn}(S_n)$ isomorphic to $S_n$ for $n\ge3$.
To do this, if I define some isomorphic function say $\phi$, where $\phi: S_n \to{\rm Inn}(S_n)$, then show that $\phi$ is bijective (by showing injective and surjective), and a homomorphism, is this enough to prove the statement above?
I show it won't hold for $S_2$, but I'm not sure if my proof is strong enough to prove isomorphism for all $n\ge3$.
Best Answer
The map $j$ that sends
$a\in S_n$
to
$f_a:S_n\to S_n$ defined by $f_a(x)=axa^{-1}$
is a group homomorphism from $S_n$ to $\text{Inn}(S_n)$.
The question is whether $j$ is bijective.
It is surjective, almost by definition.
For $n\geq3$, it is injective. Why?
$f_a=f_b$
$\implies\forall x\in S_n:axa^{-1}=bxb^{-1}$
$\implies\forall x\in S_n:x=a^{-1}\;b\;x\;b^{-1}\;a$
$\implies\forall x\in S_n:x=(b^{-1}a)^{-1}\;x\;b^{-1}a$
$\implies b^{-1}a$ commutes with every element of $S_n$
$\implies b^{-1}a=e$
$\implies a=b$.