I was reading about abelian groups but confused in Fundamental Theorem of Finitely Generated Abelian Groups, so I searched

some sources and become even more confused due to the different types of definition found.

1)

(from Abstract algebra Dummit and Foote)

**Theorem.** (*Fundamental Theorem of Finitely Generated Abelian Groups*) Let $G$ be a finitely generated abelian group. Then

**(1)** $G cong mathbb{Z}^r times Z_{n_1} times Z_{n_2} times … times Z_{n_s}$, for some integers $r$, $n_1$, $n_2$, … , $n_s$ satisfying the following conditions: **(a)** $r ge 0$ and $n_j ge 2$ for all $j$, and **(b)** $n_{i+1} mid n_i$ for $1 le i le s-1$

**(2)** the expression in (1) is unique: if $Gcong mathbb{Z}^t times Z_{m_1} times Z_{m_2} times … times Z_{m_u}$, where $t$ and $m_1$, $m_2$, … , $m_u$ satisfy (a) and (b) (i.e., $t ge 0$, $m_j ge 2$ for all $j$ and $m_{i+1} mid m_i$ for $1 le i le u-1$), then $t = r$, $u = s$ and $m_i = n_i$ for all $i$.

$:::2.$

(I can see that second definition is not for isomorphism but still,)

$$A = mathbb{Z}^n oplus mathbb{Z}_{h_1} oplus … oplus mathbb{Z}_{h_n}$$

$$where ::h_i | h_{i+1}$$

$: ; :3.$

To use the theorem you must factor suppose $vert G vert=n$ and factor $n$ into prime factors. Then you must get divisors $d_1,d_2,dots,d_m$ such that $d_imid d_{i+1}$, and $d_1cdotdotscdot d_{m}=n$, and your group can be

$$Gcong mathbb{Z}_{d_1}oplus mathbb{Z}_{d_2}oplus dots oplus mathbb{Z}_{d_m} $$

I looked in Wikipedia but they only say about uniqueness but don’t have these definitions with divisibility.

Can someone please explain why there is difference in the divisibility conditions in these definition and what each means.

I am new to these so please forgive my negligence.