I want to prove that $Aut(S_4) \cong S_4$. I saw $\operatorname{Aut}(S_4)$ is isomorphic to $S_4$ but i was troubled.

I know $Z(S_4)=1$, and so $S_4 \cong Inn(S_4)$. Thus it is sufficien to show that every automorphism of $S_4$ is inner.

Now suppose that ${ \rm Syl}_3(S_4)= \lbrace P_1,P_2, P_3, P_4 \rbrace$. Thus $Aut(S_4)$ acts on ${ \rm Syl}_3(S_4)$.

$\textbf{Watson}$ say that this action is as follow: $(P_i,f)=f(P_i)=P_{\sigma_f(i)}$. I still dont understan this action. Also why every automorphism of $S_4$ is inner?

  • $\begingroup$ ${\rm Aut}(S_4)$ acts on ${ \rm Syl}_3(S_4)$ via $f \bullet P_j := f(P_j) = P_{\sigma_f(j)} \in { \rm Syl}_3(S_4)$ for every $3$-Sylow subgroup $P_j ≤ S_4$. $\endgroup$ – Watson Sep 7 '16 at 9:01
  • $\begingroup$ I know but $f:S_4 \longrightarrow S_4$ how defined? $\endgroup$ – Hana Sep 7 '16 at 9:04
  • $\begingroup$ Here $f$ is any automorphism of $S_4$. Any automorphism of $S_4$ will induce a permutation of the $4$ $3$-Sylow subgroups of $S_4$, i.e. a permutation of $\{P_1,P_2,P_3,P_4\}$, i.e. an element of $S_4$, which I denoted by $\sigma_f$. $\endgroup$ – Watson Sep 7 '16 at 9:04
  • $\begingroup$ So $\phi : f \mapsto \sigma_f$ is a map from ${\rm Aut}(S_4) \to S_4$, which can be verified to be an homomorphism. Does this answer your question? $\endgroup$ – Watson Sep 7 '16 at 9:09
  • $\begingroup$ While $f:S_4 \to S_4$ is an automorphism, $\sigma_f$ is an element of $S_4$ which maps $j$ to $\sigma_f(j)$ such that $f(P_j) = P_{\sigma_f(j)}$. $\endgroup$ – Watson Sep 7 '16 at 9:12

To move this old question out of the unanswered list, here's an argument which works up to $n=5$.

In general, given a group $G$, the group $\operatorname{Aut}(G)$ acts on the set of the conjugacy classes of $G$, $\mathscr{C}:=\{\operatorname{cl}(a), a\in G\}$, via $\varphi\cdot \operatorname{cl}(a):=\varphi(\operatorname{cl}(a))$. Essentially, the proof that this is indeed an action boils down to proving that $\varphi(\operatorname{cl}(a))=\operatorname{cl}(\varphi(a))$. Since inner automorphisms fix each conjugacy class, we have that $\operatorname{Inn}(G)\le\operatorname{ker}\phi$, where $\phi\colon\operatorname{Aut}(G)\to S_\mathscr{C}$ is the homomorphism equivalent to the said action.

Now, if $G=S_n$, then $\operatorname{Inn}(G)=\operatorname{ker}\phi$, because the stabilizer of the conjugacy class of the transpositions is precisely equal to $\operatorname{Inn}(S_n)$ (see e.g. here). Therefore (first homomorphism theorem), $\operatorname{Aut}(S_n)/\operatorname{Inn}(S_n)\cong\phi(\operatorname{Aut}(S_n))$. If distinct conjugacy classes are "tagged" by the distinct orders of their elements, then the action is trivial, because any automorphism (order-preserving) is then class-preserving, and finally $\operatorname{Aut}(S_n)=\operatorname{Inn}(S_n)$. This happens for $n=3$, being the three distinct conjugacy classes of $S_3$ made of elements of order $1$ (the identity), $2$ (the three $2$-cycles) and $3$ (the two $3$-cycles). Therefore $\operatorname{Aut}(S_3)=\operatorname{Inn}(S_3)\cong S_3$. By relaxing such a "strong" sufficient condition ("each conjugacy class, an order of its elements"), we can encompass $n=4$ and $n=5$ cases, too. In fact, if $S_n$ has two conjugacy classes of elements of the same order, but their sizes are different, than again all the automorphisms are class-preserving (being $|\varphi(\operatorname{cl}(\sigma))|=|\operatorname{cl}(\sigma)|$), and hence again the action is trivial (whence again $\operatorname{Aut}(S_n)=\operatorname{Inn}(S_n)$). For $n=4$, the conjugacy classes of the transpositions and of the double transpositions (both whose elements have order $2$) have sizes $6$ and $3$, respectively, and hence no one automorphism can "swap" each other. Likewise, for $n=5$, the conjugacy classes of the transpositions and of the double transpositions have sizes $10$ and $15$, respectively, and the same conclusion as for $n=4$ does hold.

This simple approach can't support any further in the $n=6$ case, where we have conjugacy classes of elements of the same order, which have also the same size.


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