Finding torsion coefficients for $\mathbb{Z}_4\times\mathbb{Z}_9$

Question

The part of the decomposition theorem $11.12$ corresponding to the subgroups of prime power order can also be written in the form

$\mathbb{Z}_{m_1}\times \mathbb{Z}_{m_2}\times \cdots \times \mathbb{Z}_m$, where $m_i$ divides $m_{i+1}$ for $i=1,2,\cdots,r-1$

The numbers $m_i$ can be shown to be unique, and are the torsion coefficients of $G$

-Find the torsion coefficents of $\mathbb{Z}_4\times\mathbb{Z}_9$

The theorem $11.12$ is concerned about a finitely generated abelian group being isomorphic to

$$\mathbb{Z}_{(p_1)^{r_1}}\times \mathbb{Z}_{(p_2)^{r_2}}\times \cdots \times \mathbb{Z}_{(p_n)^{r_n}}\times \mathbb{Z}\times \mathbb{Z} \cdots \times \mathbb{Z}$$

How can I represent $\mathbb{Z}_4\times\mathbb{Z}_9$ in the form $\mathbb{Z}_{m_1}\times \mathbb{Z}_{m_2}\times \cdots \times \mathbb{Z}_m$?

Answer 1

Since $(4,9)=1$, $\Bbb{Z}_4\times \Bbb{Z}_9\cong \Bbb{Z}_{36}$ and also $\Bbb{Z}_4$ and $\Bbb{Z}_9$ cannot be decomposed into cyclic factors of smaller size.
Hence $36$ is the torsion coefficient.

Theorem:

$(m,n)=1$ iff $\Bbb{Z}_m\times \Bbb{Z}_n\cong \Bbb{Z}_{mn}$

Remark:
The torsion coefficient is commonly known as invariant factor. You can try to refer Abstract Algebra, Dummit & Foote, Section 5.2 for more informations about this.