 To see this, let be the trivial homomorphism (i.e., sending every element of to the identity automorphism of ) then is the direct product .A version of the splitting lemma for groups states that a group is isomorphic to a semidirect product of the two groups and if and only if there exists a short exact sequence

and a group homomorphism such that , the identity map on .In this case, is given by , where

The dihedral group with elements is isomorphic to a semidirect product of the cyclic groups and .