Internal regular product

Definition with symbols
Let $$H_1$$ and $$H_2$$ be subgroups of $$G$$ whose normal closures in $$G$$ are $$N_1$$ and $$N_2$$ respectively. Then $$G$$ is termed an internal regular product  of $$H_1$$ and $$H_2$$ if the following are true:


 * $$H_1$$ and $$H_2$$ generate $$G$$
 * $$H_1 \cap N_2$$ is trivial, and $$H_2 \cap N_1$$ is trivial

Relation with semidirect product
If $$G$$ is an internal regular product of subgroups $$H_1$$ and $$H_2$$ with normal closures $$N_1$$ and $$N_2$$ respectively, then $$G$$ is the semidirect product of $$N_1$$ with $$H_2$$, and is also the semidirect product of $$N_2$$ with $$H_1$$.

However, it is not necessary that every semidirect product can be viewed as coming from a regular product.

Stronger product notions

 * Direct product
 * Free product
 * Verbal product