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
The subgroup structure mirrors that of Z12: there are unique subgroups of orders 1, 2, 3, 4,
Z3Z4 serves as a standard example in the study of finite abelian groups and modular arithmetic. It