Prove that the product of two infinite cyclic groups is not inifinite cyclic.
could anyone help me to prove this ?? please?
An infinite cyclic group is isomorphic to a group of integers and the product of two infinite cyclic group is isomorphic to . First, Z and Z \timex Z are not isomorphic. The latter is not generated by a single element. If it were generated by a single element, it has the cardinality of a set of natural numbers, but it is not.
Right, a strange diagonal argument popped up in my mind when I saw this problem. I mixed that up with .
A finite product of countable sets is countable.
For instance, there is an injective map from to such that by the equation . If a set has an injective map to , it is countable.