Order of direct product is product of orders

For two groups
Suppose $$G_1$$ and $$G_2$$ are groups of orders $$a_1,a_2$$ respectively. Then, the external direct product $$G_1 \times G_2$$ has order equal to the product $$a_1a_2$$.

When $$G_1$$ and $$G_2$$ are finite groups, so is $$G_1 \times G_2$$ and all the orders are natural numbers. When either of $$G_1$$ and $$G_2$$ is an infinite group, $$G_1 \times G_2$$ is also infinite, and the statement is interpreted in terms of infinite cardinals.

For finitely many groups
Suppose $$G_1,G_2,\dots,G_n$$ are groups of orders $$a_1,a_2,\dots,a_n$$ respectively. Then, the external direct product $$G_1 \times G_2 \times \dots \times G_n$$ has order equal to the product $$a_1a_2 \dots a_n$$ or $$\prod_{i=1}^n a_i$$.

When all the groups $$G_i$$ 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 fact about::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 $$I$$ is an indexing set and $$G_i, i \in I$$ is a collection of groups. Then, the order of the (unrestricted) external direct product of the $$G_i$$s is the (possibly infinite) product of the orders of the $$G_i$$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.

Similar facts

 * Order of semidirect product is product of orders
 * Order of extension group is product of order of normal subgroup and quotient group