Outer Semidirect Product
Given any two groups and and a group homomorphism , we can construct a new group , called the (outer) semidirect product of and with respect to , defined as follows.
(i) The underlying set is the Cartesian product .
(ii) The operation, , is determined by the homomorphism :
for and .
This defines a group in which the identity element is and the inverse of the element is .
Pairs form a normal subgroup isomorphic to , while pairs form a subgroup isomorphic to .
Inner Semidirect Product (Definition)
Given a group with identity element , a subgroup , and a normal subgroup ; then the following statements are equivalent:
(i) is the product of subgroups, , where the subgroups have trivial intersection, .
(ii) For every , there are unique and , such that .
If these statements hold, we define to be the semidirect product of and , written .
Inner Semidirect Product Implies Outer Semidirect Product
Suppose we have a group with , and every element can be written uniquely as where , .
Define as the homomorphism given by , where for all , .
Then is isomorphic to the semidirect product , and applying the isomorphism to the product, , gives the tuple, . In , we have
which shows that the above map is indeed an isomorphism.