# Order of semidirect product is product of orders

This article gives an expression for the value of the arithmetic function order of a group of a group obtained by applying a group operation external semidirect product in terms of the values for the input groups. It says that the value for the group obtained after performing the operation is the product of the values for the input groups.
View facts about order of a group: (facts closely related to order of a group, all facts related to order of a group)
View facts about external semidirect product: (facts closely related to external semidirect product, all facts related to external semidirect product)
View facts about product: (facts closely related to product, all facts related to product)

## Statement

### For external semidirect product

Suppose $N$ and $H$ are groups, $\rho: H \to \operatorname{Aut}(N)$ is a homomorphism of groups, and $G = N \rtimes H$ is the external semidirect product of $N$ by $H$ for the action $\rho$.

Then, the order of $G$ is the product (in the sense of multiplication) of the order of $N$ and the order of $H$.

When both $N$ and $H$ are finite groups, so is $G$ and the above statement is true in the sense of multiplication of finite numbers. When either $N$ or $H$ is an infinite group, so is $G$, and the above statement is true in the sense of multiplication of cardinals.

### For internal semidirect product

Suppose $G$ is a group, $N$ is a complemented normal subgroup of $G$, and $H$ is a complement to $N$ in $G$. Thus, $G$ is the internal semidirect product of $N$ and $H$.

Then, the order of $G$ is the product of the order of $N$ and the order of $H$.