About the construction of semidirect products

Question

I need help with the following question:

We have $G_0=H_0[N]_\rho$ a semidirect product, and $\pi:H\to H_0$ an epimorphism such that $K=\ker(\pi)\unlhd H$, and $H/K\cong H_0$.

We have to construct a natural semidirect product $G=H[N]_{\rho'}$, and check that $K\unlhd G$, and $G/K\cong G_0$.

Finally, using this method, we have to construct groups of the form $C_4[C_3]$ and $Q_8[C_3]$, starting from $G_0=S_3$.

Thanks a lot in advance for any help!

Answer 1

By definition $G_0=N\rtimes_{\rho}H_0$, where $\rho:H_0\to Aut(N)$ is a group homomorphism. You have another group homomorphism $\rho'=\rho\pi:H\to Aut(N)$ which gives rise to a semidirect product $G=N\rtimes_{\rho'}H$. Define $\varphi:G\to G_0$ by $\varphi(n,h)=(n,\pi(h))$ and show that $\varphi$ is a surjective group homomorphism with $\ker\varphi=\{1_N\}\times K$, so $K\trianglelefteq G$ and $G/K\cong G_0$.

If $G_0=S_3$, then $G_0=C_3\rtimes_{\rho} C_2$, where $\rho:C_2\to Aut(C_3)$ is the only non-trivial homomorphism. Now all you have to do is to choose a surjective group homomorphism $\pi:C_4\to C_2$, respectively $\pi:Q_8\to C_2$.

Answer 2

Hint: An action of $A$ on $B$ corresponds to a map $A\rightarrow \operatorname{Aut}(B)$. So we know that there exists a map $H_0\rightarrow \operatorname{Aut}(N)$. Conclude that $H$ acts on $N$, so you have the semidirect product $H\rtimes N$ you are after.