I would like to know if the statement and my proof are correct, as this seems surprising to me.
Let $G$ be a group of order 24 without normal sylow subgroups.
From sylow theory, $n_2=3$, $n_3=4$. Let $G$ act by conjugation on its sylow $3$-subgroups with the corresponding homomorphism $\phi : G \to S_4$.
The action is transitive from sylow's theorems.
Further, a sylow $3$-subgroup cannot normalize any other sylow $3$ subgroup, thus any element of a sylow $3$ subgroups acts as a 3-cycle on the sylow subgroups not containing it.
Taking the square of such an element results in the inverse $3$ cycle. Thus, $\phi(G) \le S_4$ must contain all $3$-cycles in $S_4$, so it contains $A_4$. It is left to prove that it equals $S_4$.
Assume by contradiction that $\phi(G) = A_4$. then $\phi$ has a kernel of order 2. So there is a single nontrivial element in the kernel, $z$. Since $\left< z \right>$ is the kernel of $\phi$, it is normal in $G$. Consider $G/\left<z\right>$: Since $\left< z \right>$ is normal in $G$, it is contained in all three $2$-sylow subgroups of $G$. From the lattice isomorphism theorem, we can conclude that there are three $2$-sylow subgroups of $G/ \left< z \right>$.
It is simple to prove that all groups of order 12 must have some normal sylow subgroup (via counting elements). Thus, $G/ \left< z \right>$ must have a normal sylow $3$-group. Thus the images of the four $3$-sylow subgroups of $G$ under the natural projection must be equal. Thus, there are four distinct elements of $G$ whose class in $G/ \left< z \right>$ is equal, which is absurd as $ | \left< z \right> | = 2$.
Is my proof correct? Alternative proofs are welcome.
