1. Semidirect Product and Its Properties
Definition A.1.1. Let G_{1} and G_{2}
be the groups and
f: G_{1} ®
Aut(G_{2}) the homomorphism between these two groups,
where Aut(G_{2}) is the set of all automorphisms
of the group G_{2}. The semidirect product of G_{1}
and G_{2} with the operation f
is
G_{1}×_{f}G_{2}
= {(g_{1},g_{2})  g_{1}Î
G_{1}, g_{2} Î
G_{2} }, 

where
(g_{1},g_{2})(g_{1}¢,g_{2}¢)
= (g_{1}f(g_{2})(g_{1}¢),g_{2}g_{2}¢). 

The identity is (e_{1},e_{2}), where e_{1}
is the identity element from G_{1}, and e_{2}
the identity from G_{2}. The inverse element of (g_{1},g_{2})
is (f(g_{2})^{1}(g_{1}^{1}),g_{2}^{1}).
From
[(g_{1},g_{2})(g_{1}¢,g_{2}¢)]
(g_{1}¢¢,g_{2}¢¢) 

= (g_{1}f(g_{2})(g_{1}¢),g_{2}g_{2}¢)(g_{1}¢¢,g_{2}¢¢) 


= (g_{1}f(g_{2})(g_{1}¢)f(g_{2}g_{2}¢)(g_{1}¢¢),g_{2}g_{2}¢g_{2}¢¢)
=f (g_{1}f(g_{2})(g_{1}¢f(g_{2}¢)(g_{1}¢¢)),g_{2}g_{2}¢g_{2}¢¢) 


= (g_{1},g_{2})(g_{1}¢f(g_{2}¢)(g_{1}¢¢),g_{2}¢g_{2}¢¢) 

= (g_{1},g_{2})[(g_{1}¢,g_{2}¢)(g_{1}¢¢,g_{2}¢¢)]
, 
