I'm reading A First Course in Abstract Algebra by Fraleigh and I've reached a point where I feel like I'm supposed to have understood something more from the chapter than what is actually stated. I've done (most of) the exercises leading up to the question and I've reread everything at least 5 times.
Why is $\mathbb{Z}_8\times\mathbb{Z}_{10}\times\mathbb{Z}_{24}$ not isomorphic to $\mathbb{Z}_4\times\mathbb{Z}_{12}\times\mathbb{Z}_{40}$?
The fundamental theorem of finitely generated abelian groups (FToFGAG) says that every FGAG is isomorphic to a direct product of cyclic groups in the form:
$\mathbb{Z}_{p_1^{r_1}}\times\mathbb{Z}_{p_2^{r_2}}\times\ldots\times\mathbb{Z}_{p_n^{r_n}}\times\mathbb{Z}\times\ldots\times\mathbb{Z}$.
That is what the book states. Okay. Maybe I'm really tired, but it seems like, by the theorem, they should both be isomorphic to $\mathbb{Z}_{2^7}\times\mathbb{Z}_{5}\times\mathbb{Z}_{3}$.
Why is that not true?