Editing Isomorphism (section)

===Integers modulo 6===
Consider the group <math>(\Z_6, +),</math> the integers from 0 to 5 with addition [[Modular arithmetic|modulo]]&nbsp;6. Also consider the group <math>\left(\Z_2 \times \Z_3, +\right),</math> 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:
<math display="block">\begin{alignat}{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{alignat}</math>
or in general <math>(a, b) \mapsto (3 a + 4 b) \mod 6.</math>

For example, <math>(1, 1) + (1, 0) = (0, 1),</math> which translates in the other system as <math>1 + 3 = 4.</math>

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 groups|direct product]] of two [[cyclic group]]s <math>\Z_m</math> and <math>\Z_n</math> is isomorphic to <math>(\Z_{mn}, +)</math> if and only if ''m'' and ''n'' are [[coprime]], per the [[Chinese remainder theorem]].