Currently i am trying to understand the concept of Semi direct product of groups from Abstract Algebra text of Dummit and Foote.The discussion given in the same book is bit confusing to me and i am unable to understand this concept properly.
What is the difference between internal and external semi direct product ? I found different definitions one by using the group action and other by using the splitting of exact sequence but i could not figure out how these definitions are equivalent.
Definition: Consider $G$,$H$ groups and $\varphi : G\to \operatorname{Aut}(H)$ a homomorphism, we define the semidirect product group as the cartesian product $G\times H$ together with the operation
$$(g_1,h_1)(g_2,h_2)=(g_1g_2,h_1\varphi(g_1)(h_2)),$$
and we denote the resulting group as $G\ltimes H$.
At some places i found the following definition also:
Definition: Existence of a split short exact sequence $1 \rightarrow N \rightarrow G \rightarrow H \rightarrow 1$ gives that $G$ is isomorphic to a semidirect of $N$ and $H$
Please suggest any book/notes to understand this concept properly.
Thank you