Cp×q

Cp×q denotes the direct product of two cyclic groups: the cyclic group of order p and the cyclic group of order q. In common notation this is written as C_p × C_q, and it is an abelian group of order pq.

A key property is whether the product is cyclic. Cp×q is cyclic if and only if gcd(p, q) = 1. In particular, if p and q are distinct primes, Cp×q is isomorphic to the cyclic group of order pq, C_{pq}, by the Chinese remainder theorem. Equivalently, there exists an element of order pq in Cp×q when p and q are coprime.

If gcd(p, q) > 1, Cp×q is not cyclic. For example, Cp×Cp with p prime has order p^2 and is not cyclic; it is a two-dimensional vector space over the field with p elements, often described as an elementary abelian p-group of rank 2. In general, when p and q share a common factor, the exponent of Cp×q is lcm(p, q) and the group cannot be generated by a single element.

The structure of Cp×q is a standard example in the study of finite abelian groups. It demonstrates how direct products can yield either cyclic groups (when the orders are coprime) or non-cyclic abelian groups (when they are not). The group has p×q elements, and its elements can be described as ordered pairs (a, b) with a in C_p and b in C_q, with operation defined componentwise.

See also: direct product, cyclic group, Chinese remainder theorem, finite abelian group.