In this post, we study the Fundamental Theorem of Finitely Generated Abelian Groups, and as an application we solve the following problem.

Problem.
Let $G$ be a finite abelian group of order $n$.
If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic to the cyclic group $Z_n=\Zmod{n}$ of order $n$.

Fundamental Theorem of Finitely Generated Abelian Groups

Before stating the fundamental theorem for finitely generated abelian groups, we define several terminologies and notations.

Definitions / notations

• We say that a group $G$ is finitely generated if there is a finite subset $S$ of $G$ such that $G$ is generated by $S$, that is, $G=\langle S \rangle$.
• For each positive integer $r$, let
  \[\Z^r=\underbrace{\Z\times \Z \times \cdots \times \Z}_{\text{$r$ times}}\] be the direct product of $r$ copies of $\Z$. Here we set $\Z^0=1$ to be the trivial group.
• The group $\Z^r$ is called the free abelian group of rank $r$.
• For each positive integer $n$, let $\Z_n=\Zmod{n}$ be the cyclic group of order $n$.

Theorem (Fundamental Theorem of Finitely Generated Abelian Groups)

• The integer $r$ in the decomposition (*) is called the free rank or Betti number of $G$.
• The integers $n_1, n_2, \dots, n_s$ are called the invariant factors of $G$.
• The decomposition (*) is called the invariant factor decomposition of $G$.

Problem

Let $G$ be a finite abelian group of order $n$.
If $n$ is the product of distinct prime numbers, then prove that $G$ is isomorphic to the cyclic group $Z_n=\Zmod{n}$ of order $n$.

Proof.

Since $G$ is a finite abelian group, it is in particular a finitely generated abelian group.
(We can take $G$ itself for a finite set of generators of $G$.)
Then by the fundamental theorem of finitely generated abelian groups, we have the invariant factor decomposition
\[G\cong \Z^r\times Z_{n_1}\times \Z_{n_2}\times \cdots \times \Z_{n_s}\] satisfying

1. $r\geq 0$ and $n_i \geq 2$ for all $i$, and
2. $n_{i+1}|n_i$ for $1 \leq i \leq s-1$.

Since $G$ is a finite group, the rank $r$ must be $0$. Thus we have an isomorphism
\[G\cong Z_{n_1}\times \Z_{n_2}\times \cdots \times \Z_{n_s}.\] Comparing the order, we have
\[n=n_1 n_2\cdots n_s.\]

Let $p$ be a prime factor of $n$. Then $p$ divides some $n_i$.
If $i>1$, then it follows from condition (2) that $p$ divides $n_1$ as well.
Thus $p^2$ divides $n$. Since $n$ is a square-free integer, this is a contradiction.
It follows that any prime factor of $n$ divides only $n_1$.

Therefore we obtain $n=n_1$ and $s=1$. So the invariant factor decomposition of $G$ is
\[G\cong Z_n.\] Hence $G$ is isomorphic to the cyclic group $\Z_n$ of order $n$.

Reference

Abstract Algebra by Dummit and Foote (third edition) Section 5.2.

The post Fundamental Theorem of Finitely Generated Abelian Groups and its application first appeared on Problems in Mathematics.