Help understanding Fundamental Theorem of Finite Abelian Groups The book I'm reading defines the Fundamental Theorem of Finite Abelian Groups in the usual way that a finite abelian group is isomorphic to a direct product of cyclic groups $\mathbb{Z}_{{p_1}^{e_1}} \times \cdots \times  \mathbb{Z}_{{p_r}^{e_r}}$, but then it also states that the group is isomorphic to $\mathbb{Z}_{m_1} \times \cdots \times  \mathbb{Z}_{m_t}$ where for each $i = 1,...,t-1$ we have $m_i | m_{i+1}$. How can this be if the groups orders are supposed to be relatively prime.
If anyone can give an example of how this works I would greatly appreciate it.
 A: That is because of the Chinese remainder theorem. If the group orders are pairwise coprime, the product is isomorphic to $\;\mathbf Z/p_1^{e_1}\dots p_r^{e_r}\mathbf Z$, and there is only one such $m_i$ (which, incidentally, are called the invariant factors of the abelian group).
Here is an example:  suppose the primary decomposition of the group is:
$$\mathbf Z/2\mathbf Z\times \mathbf Z/3^2\mathbf Z\times \mathbf Z/3^2\mathbf Z\times \mathbf Z/5\times\mathbf Z/5^2\mathbf Z\times  \mathbf Z/5^3\mathbf Z. $$
First display the primary factors on different lines (one line per prime factor) in increasing order of the exponent and right-aligned:
 $$\begin{array}{r}
\mathbf Z/2\mathbf Z^{\phantom{2}}\\
 {}\times \mathbf Z/3^2\mathbf Z\times \mathbf Z/3^2\mathbf Z\\
  {}\times \mathbf Z/5\mathbf Z\times\mathbf Z/5^2\mathbf Z\times  \mathbf Z/5^3\mathbf Z
\end{array}.  $$
Then apply the C.r.t. to each column:
$$ \mathbf Z/5\mathbf Z\times \mathbf Z/225\mathbf Z\times  \mathbf Z/2250\mathbf Z.$$
