Non-isomorphic abelian groups of order 8

Question

$\newcommand{\Z}{\mathbb Z}$

I don't understand why the following groups are not isomorphic with each other $\Z/8\Z, $ $\Z/4\Z \times \Z/2\Z,$ and $(\Z/2\Z)^3$.

Indeed, i thought because they re all abelian finite groups with order equals $8$. So they were isomorphic.

thanks

Answer 1

The group $\mathbb{Z}/8\mathbb{Z}$ has an element of order $8$.

The group $(\mathbb{Z}/4\mathbb{Z})\times(\mathbb{Z}/2\mathbb{Z})$ has an element of order $4$ but no element of order $8$.

The group $(\mathbb{Z}/2\mathbb{Z})^3$ has no element whose order is greater than $2$.