Let $G$ a group and $N$ a normal subgroup of $G$. If $G$ it have a subgroup $H$ s.t. $H \cap N $ is the trivial subgroup and $H$ is isomorphic to $G/ N$ then $G$ is isomorphic to $N\rtimes H$.
Could someone help me to understand the general idea and motivation of this construction sorry but at this moment seems totally bizarre, jaja ?
I have an idea but I'm not sure if we show that the composition of the canonical map from $G\to G/N$ and the isomorphism $G/N$ to $H$ which we call $f$ is the identity under the elements in $H$ and is kernel is $N$, so we can write the elements in $G$ as product on the form $G= NH$ and from here construct the homomorphism from $H\to Aut (N)$ and hope that this give us the isomorphism, my intuition is slightly correct?
I have to go, later on write down the details...