z1z2

Z1Z2 is the direct product Z1 × Z2, where Zn denotes the cyclic group of order n (the integers modulo n under addition). The group Z1 is the trivial group with a single element, typically 0. Z2 consists of two elements, 0 and 1, with 1 + 1 = 0. The direct product Z1 × Z2 consists of ordered pairs (a,b) with a ∈ Z1 and b ∈ Z2, with componentwise addition. Since Z1 has only one element, every element of Z1 × Z2 has the form (0,b), so Z1 × Z2 is isomorphic to Z2 via the map (0,b) ↦ b. Therefore Z1Z2 is a cyclic group of order 2.

As a member of the family Zm × Zn, the group Zm × Zn is cyclic of order mn if gcd(m,n) = 1; here gcd(1,2) = 1, so Z1 × Z2 ≅ Z2. Consequently it shares the same structure as Z2 and is the simplest nontrivial finite abelian group. Its elements can be listed as (0,0) and (0,1), and any nonzero element generates the group.

In mathematical contexts, Z1Z2 serves as a basic example of a direct product decomposition and a building block in the classification of finite abelian groups. It also illustrates how a trivial factor can be absorbed into a nontrivial one in direct products.