Product of cardinals

Product of two cardinals
The product of two cardinals $$\alpha,\beta$$ is defined as the cardinality of any set obtained as the Cartesian product of a set of cardinality $$\alpha$$ with a set of cardinality $$\beta$$. The product may be denoted $$\alpha\beta$$.

Product of finitely many cardinals
Suppose $$\alpha_1,\alpha_2,\alpha_3,\dots,\alpha_n$$ are cardinals (possibly equal, possibly distinct). The product cardinal, denoted $$\alpha_1\alpha \dots \alpha_n$$ is the cardinality of any set that is expressed as the Cartesian product of sets $$A_1,A_2,\dots,A_n$$ where the cardinality of $$A_i$$ is $$\alpha_i$$ for each $$i \in \{ 1,2,\dots,n\}$$.

We can also define this iteratively by multiplying the cardinals two at a time. Cardinal multiplication is commutative and associative, so the manner of parenthesization and order of multiplication do not matter.

Computation of product
We list the following simple rules to compute products of cardinals:

Note that if the cardinals are arising as orders of groups, we do not need to worry about any of them being zero.

Computation of one factor given product and other factor
We consider the case of a product of two cardinals.

Related notions

 * Product of infinitely many cardinals generalizes this to products of infinite length, where the cardinals individually may be finite or infinite.
 * Restricted product of infinitely many cardinals