Give all groups of cardinality 12.
What I did:
let G be such a group.
$12 = 2^2.3$ so G has a 2-Sylows $S_2$ and a 3-Sylows $S_3$.
I proved that either $S_2$ or $S_3$ is normal and $S_3\cap S_2=\{e\}$. Therefore: $G\cong S_2\rtimes S_3$ or $G\cong S_3\rtimes S_2$.
We have $S_2\cong \Bbb Z/4\Bbb Z$ or $S_2 \cong (\Bbb Z/2\Bbb Z)^2$ and $Aut(S_3) = Aut(\Bbb Z/4\Bbb Z)=\Bbb Z/2\Bbb Z$.
- Using Chinese lemma, I found the following direct products, $\Bbb Z/4\Bbb Z \times \Bbb Z/3\Bbb Z\cong \Bbb Z/12\Bbb Z,\ \Bbb Z/2\Bbb Z \times \Bbb Z/6\Bbb Z,\ \Bbb Z/2\Bbb Z \times \Bbb Z/2\Bbb Z \times \Bbb Z/3\Bbb Z$.
- I don't know how to compute $Aut((\Bbb Z/2\Bbb Z)^2)$ in order to find non trivial morphisms $S_3 \to Aut((\Bbb Z/2\Bbb Z)^2)$ and I have trouble finding non trivial morphisms $S3 \to Aut(S_2)$ and $S_2 \to Aut(S_3)$ in order to find the semidirect products.
Thank you for your help and comments