Internal regular product

This article describes a product notion for groups. See other related product notions for groups.

Definition

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 other properties

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