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 $\displaystyle \mathbb{Z}$ and the product of two infinite cyclic group is isomorphic to $\displaystyle \mathbb{Z} \times \mathbb{Z}$. 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 $\displaystyle \{0,1\}^\omega$.
A finite product of countable sets is countable.
For instance, there is an injective map from $\displaystyle \mathbb{Z}_+ \times \mathbb{Z}_+$ to $\displaystyle \mathbb{Z}_+ $ such that$\displaystyle f:\mathbb{Z}_+ \times \mathbb{Z}_+ \rightarrow \mathbb{Z}_+ $ by the equation $\displaystyle f(n,m) = 2^n3^m$. If a set has an injective map to $\displaystyle \mathbb{Z}_+$, it is countable.