I am stuck in this homework question.
Please help me to find the order of each of the elements in the direct product of $Z_3 \times Z_6$.
This is what I got so far.
- elements of $Z_3 = \{0,1,2\}$
- elements of $Z_6 = \{0,1,2,3,4,5\}$
$$Z_3 \times Z_6= \{(0,0), (0,1), (0,2), (0,3)(0,4),(0,5),(1,0), (1,1), (1,2), (1,3)(1,4),(1,5),(2,0), (2,1), (2,2), (2,3)(2,4),(2,5)\}$$
Please help me to understand and apply the theorem to answer this questions.
Let (a1, a2, . . . , an) ∈ Qn
i=1 Gi. If ai is of finite order ri in Gi, then the order of (a1, a2, . . . , an) in Qn i=1 Gi is equal to the least common multiple of all the ri.
Hint: The order of an element $(a,b)\in G\times H$ is $\rm{lcm}(\rm{ord}(a),\rm{ord}(b))$.