A "split" extension of groups and which contains a subgroup isomorphic to with and (Ito 1987, p. 710). Then the semidirect product of a group by a group , denoted (or sometimes ) with homomorphism is given by
where , , and (Suzuki 1982, p. 67; Scott 1987, p. 213). Note that the semidirect product of two groups is not uniquely defined.
The semidirect product of a group by a group can also be defined as a group which is the product of its subgroups and , where is normal in and . If is also normal in , then the semidirect product becomes a group direct product (Shmel'kin 1988, p. 247).