How do I show every abelian group whose order is square free is cyclic without using the fundamental theorem of finite abelian groups?
I tried something like this Let $|G| = p_1p_2...p_n$ By Cauchy's theorem, there's an element $x_i$ of order of $p_i$. Now, I want to show that $x_1x_2...x_n$ generates G. Is this approach correct?