semidirect product problem

Question

I am trying to show that the semidirect product of $G$ with $G$, where $G$ is a finite group, with the automorphism by conjugation on itself, is isomorphic to direct product of $G$ with $G$. Please help

Answer 1

In $G \rtimes G$, we have $$ (g_1,g_2)\cdot(h_1,h_2) = (g_1 h_1^{g_2}, g_2 h_2). $$ Use this operation and the action $h^g=ghg^{-1}$ to prove the following:

Let $H =\{(g, 1) \mid g\in G\} \leq G \rtimes G$. We then have $H \trianglelefteq G \rtimes G$ and $H \cong G$. Next, let $K=\{(g, g^{-1}) \mid g\in G\} \leq G \rtimes G$ and show $K \trianglelefteq G \rtimes G$ and $K \cong G$. Finally, note $H \cap K = 1$ and show that $HK = G \rtimes G$. This gives $G \rtimes G \cong G \times G$ as desired.