Im working on my thesis about semidirect products and splitting lemma. I got the following theorems to prove and Im a not sure how to start. I would appreciate any help.
$\\$ 1. Let $f:A\to B$ be a map.
Show:
a) if $g:B\to A$ so that $gf=id_{A}$ then $f$ is injective
b) if $g:B\to A$ so that $fg=id_{B}$ then $f$ is surjective
$\\$
$A$, $B$, $G$ groups and there is a short exact sequence
$1\to A\to G\to B\to 1$
then $\alpha :A\to G$ is injective and $\beta :G\to B$ is surjective. Please show that.