Z3Z4

Z3Z4 denotes the direct product Z3 × Z4, the additive group formed by pairs (a,b) where a ∈ Z3 and b ∈ Z4, with componentwise addition: (a,b) + (a',b') = (a + a' mod 3, b + b' mod 4). The group has 12 elements and is abelian.

Because gcd(3,4) = 1, Z3 × Z4 is cyclic and isomorphic to Z12. In particular, there exists an element that generates the entire group; for example, (1,1) has order 12. Elements can be listed as (a,b) with a ∈ {0,1,2} and b ∈ {0,1,2,3}. The order of an element (a,b) is lcm(ord(a), ord(b)), where ord(a) is 1 or 3 in Z3 and ord(b) is 1, 2, or 4 in Z4. Examples include (1,0) of order 3 and (0,1) of order 4.

The subgroup structure mirrors that of Z12: there are unique subgroups of orders 1, 2, 3, 4, 6, and 12. For instance, the subgroup generated by (0,2) has order 2, the one generated by (1,0) has order 3, and the one generated by (0,1) has order 4; the whole group is generated by (1,1) with order 12.

Z3Z4 serves as a standard example in the study of finite abelian groups and modular arithmetic. It illustrates the Chinese remainder theorem in action and the fact that the direct product of cyclic groups with coprime orders is itself cyclic. It is commonly denoted Z3 × Z4 or Z3 ⊕ Z4, and is isomorphic to Z12.