# Order of direct product is product of orders

This article gives an expression for the value of the arithmetic function order of a group of an external direct product in terms of the values for the direct factors. It says that the value for the direct product is the product of the values for the direct factors.

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 direct product: (factscloselyrelated to external direct product, all facts related to external direct product)

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

## Contents

## Statement

### For two groups

Suppose and are groups of orders respectively. Then, the external direct product has order equal to the product .

When and are finite groups, so is and all the orders are natural numbers. When either of and is an infinite group, is also infinite, and the statement is interpreted in terms of infinite cardinals.

### For finitely many groups

Suppose are groups of orders respectively. Then, the external direct product has order equal to the product or .

When all the groups are finite groups, so is the direct product, and all the orders are natural numbers.

When any of the groups is infinite, so is the direct product, and the statement is interpreted in terms of infinite cardinals. Note that, in this latter case, assuming the axiom of choice, the product of the infinite cardinals is the *maximum* of the infinite cardinals being multiplied.

### For internal direct products

The statement above holds if we replace external direct product by Internal direct product (?), i.e., the order of the internal direct product is the product of the orders. Note that by equivalence of internal and external direct product, the statements for both kinds of products are equivalent.

### For infinitely many groups using unrestricted direct products

Suppose is an indexing set and is a collection of groups. Then, the order of the (unrestricted) external direct product of the s is the (possibly infinite) product of the orders of the s. Note that this infinite multiplication is done according to various rules for cardinal multiplication. The rule for computing the product is as follows: it is the **maximum** of the following two things:

- The maximum of the cardinals being multiplied
- The power cardinal of (the number of cardinals strictly greater than 1 that are being multiplied)

Note, in particular, that if each of the groups is finite, then the order of the unrestricted direct product is the power cardinal of the number of groups that have order strictly greater than 1, or equivalently, the number of nontrivial factors of the product.

### For infinitely many groups using restricted direct products

Note that for infinitely many groups, the internal direct product matches, not the unrestricted external direct product, but the restricted external direct product. There is a notion of multiplication that, when fed with the orders of the groups, gives the order of the restricted external direct product. The product is defined as the **maximum** of the following two things:

- The maximum of the cardinals being multiplied
- The number of cardinals strictly greater than 1 that are being multiplied

Note, in particular, that if each of the groups is finite, then the order of the unrestricted direct product is the number of groups that have order strictly greater than 1.

## Related facts

- Lagrange's theorem states that the order of a group equals the product of the order of a subgroup and the size of its coset space. Interpreted for an internal direct product, we can take either of the direct factors as the subgroup, in which case the quotient will be isomorphic to the other direct factor. Direct products are thus a
*special illustration*of Lagrange's theorem. However, whereas Lagrange's theorem holds only for groups and not for loops or monoids, the order of a direct product is the product of the orders in all these cases as well.