# Order of semidirect product is product of orders

From Groupprops

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: (factscloselyrelated to order of a group, all facts related to order of a group)

View facts about external semidirect product: (factscloselyrelated to external semidirect product, all facts related to external semidirect product)

View facts about product: (factscloselyrelated to product, all facts related to product)

## Statement

### For external semidirect product

Suppose and are groups, is a homomorphism of groups, and is the external semidirect product of by for the action .

Then, the order of is the product (in the sense of multiplication) of the order of and the order of .

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

### For internal semidirect product

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

Then, the order of is the product of the order of and the order of .