Isomorphism
Integers modulo 6

Consider the group (Z 6, +), {\displaystyle (\mathbb {Z} _{6},+),} the integers from 0 to 5 with addition modulo 6. Also consider the group (Z 2 × Z 3, +), {\displaystyle \left(\mathbb {Z} _{2}\times \mathbb {Z} _{3},+\right),} the ordered pairs where the x coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the x -coordinate is modulo 2 and addition in the y -coordinate is modulo 3.

These structures are isomorphic under addition, under the following scheme: (0, 0) ↦ 0 (1, 1) ↦ 1 (0, 2) ↦ 2 (1, 0) ↦ 3 (0, 1) ↦ 4 (1, 2) ↦ 5 {\displaystyle {\begin{alignedat}{4}(0,0)&\mapsto 0\\(1,1)&\mapsto 1\\(0,2)&\mapsto 2\\(1,0)&\mapsto 3\\(0,1)&\mapsto 4\\(1,2)&\mapsto 5\\\end{alignedat}}} or in general (a, b) ↦ (3 a + 4 b) mod 6. {\displaystyle (a,b)\mapsto (3a+4b)\mod 6.}

For example, (1, 1) + (1, 0) = (0, 1), {\displaystyle (1,1)+(1,0)=(0,1),} which translates in the other system as 1 + 3 = 4. {\displaystyle 1+3=4.}

Even though these two groups "look" different in that the sets contain different elements, they are indeed isomorphic: their structures are exactly the same. More generally, the direct product of two cyclic groups Z m {\displaystyle \mathbb {Z} _{m}} and Z n {\displaystyle \mathbb {Z} _{n}} is isomorphic to (Z m n, +) {\displaystyle (\mathbb {Z} _{mn},+)} if and only if m and n are coprime, per the Chinese remainder theorem.
language:response_language[Latin], oulipo:avoid_letter[a]