Essentially, it is when you pin together two groups (such that they are still groups in their own right - they are both normal in the new group and their intersection is trivial), and then you say how they interact with one another. For instance, with the direct product you say that they commute with one another, while the dihedral group is the semidirect product of with such that the elements of the group interact according to the rule .
actually both are not necessarily normal but one of them will be normal. internally, we say that a group is a semidirect product of two subgroups and and we write if:
and a more general definition is to say that is a semidirect product of two groups and we write if there exists an exact sequence
which is split, i.e. there is a map such that Then it's easy to see that is an internal semidirect product of subgroups
and an equivalent definition is to assume that there is a group homomorphism then define the semidirect product of with respect to to be the set
with multiplication defined by: then we write again it's easy to see that is an internal semidirect product of two subgroups
and