Proving a criterion for recognizing when a group $G$ is a semidirect product of 2 groups

Question

Here is the question I want to prove:

For groups $G,H,K,$ show that the following conditions are equivalent.

1. $G \cong K \rtimes_{\varphi} H.$ where $\varphi : H \rightarrow Aut(K).$

2. There exists a right-split short exact sequence: $1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1.$

3. $H \subset G, K \triangleleft G, G = HK $ and $H \cap K = \{1\}.$

My questions are:

1-Is there any textbooks contains the proof of $1 \Leftrightarrow 3.$?

2- Can anyone help me in proving $1 \Rightarrow 2$?

3- Can anyone help me in proving $2 \Rightarrow 3$?

Answer 1

“Group Theory” by J.S. Milne contains a rather detailed section (from p. 46) devoted to semiderect products of groups. The next section is devoted to extensions of groups related to semidirect products. “Abstract Algebra: The Basic Graduate Year” by Robert B. Ash has a series of problems for Chapter 5.8 related to the topic, with solutions.

Answer 2

I would recommend looking at K. Conrad's SPLITTING OF SHORT EXACT SEQUENCES FOR GROUPS where I suppose you'll find what you want