# Order of extension group is product of order of normal subgroup and quotient group

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 group extension 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 group extension: (factscloselyrelated to group extension, all facts related to group extension)

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

## Contents

## Statement

Suppose is a group, is a normal subgroup, and is the corresponding quotient group. Then, we have the following relation between the orders of :

In other words, the order of the extension group for a group extension is the product of the orders of the normal subgroup and the quotient group.

Note that:

- The order of is finite if and only if both the order of and the order of are finite, and in this case the statement is interpreted using multiplication of natural numbers.
- If any of the orders is infinite, the statement is interpreted in terms of cardinal multiplication.

## Related facts

- Lagrange's theorem
- Order of quotient group divides order of group
- Order of element divides order of group

## Facts used

## Proof

The proof follows directly from Fact (1).