A pressing question:
If we are given two cyclic groups, and we take a cartesian product of the two, is the result also a cyclic group?
In fact, is cyclic iff . To think of why this is we need some element of to be of exactly order but it is clear the order of is . Therefore lastly noting that we conclude that . And since all cyclic groups of order are isomorphic to this is the case for any cyclic groups.