Z2Z3

Z2Z3 commonly denotes the direct product of the cyclic groups of orders 2 and 3, usually written Z2 × Z3 or Z/2Z × Z/3Z. It is a finite abelian group with six elements and componentwise addition modulo 2 and modulo 3. Explicitly its elements are pairs (a, b) with a in {0,1} and b in {0,1,2}, and its group operation is (a, b) + (a', b') = (a + a' mod 2, b + b' mod 3).

Because 2 and 3 are coprime, Z2 × Z3 is cyclic of order 6 and is isomorphic to the cyclic group Z6. The isomorphism can be given by the Chinese remainder theorem: each pair (a, b) corresponds to the unique residue class modulo 6 that is congruent to a modulo 2 and to b modulo 3. Element orders reflect the direct product structure: elements of form (1,0) have order 2, (0,1) have order 3, and some elements such as (1,1) have order 6 and generate the whole group.

Z2 × Z3 appears in elementary group theory and number theory as a simple example illustrating direct products, cyclicity conditions, and the Chinese remainder theorem. It also serves in classifications of finite abelian groups: more generally, Zm × Zn is isomorphic to Zmn when m and n are coprime.