I saw the following problem in a book, and am not sure how to approach it. Given the group $Z_{p^3}$, does it have a subgroup isomorphic to $Z_p \times Z_p$.
I know that it's a well known problem to prove that $Z_{p^2}$ is not isomorphic to $Z_p \times Z_p$, as the first has an element of order $p^2$, while the latter does not, but I am not sure how to solve this variant. Thanks for your help.