nLab finite abelian group

Definition

A finite abelian group is a group which is both finite and abelian.

Properties

As any finite group, a finite abelian group is pure torsion.

Proposition

If a finite abelian group $A$ has order ${\vert A \vert} = p$ a prime number, then it is the cyclic group $\mathbb{Z}_p$.

Proposition

If $A$ is a finite abelian group and $p \in \mathbb{N}$ is a prime number that divides the order ${\vert A \vert}$, then equivalently

• $A$ has an element of order $p$;

• $A$ has a subgroup of order $p$.

This is Cauchy's theorem restricted to abelian groups.

Proof

We prodeed by induction on the order of $A$. For ${\vert A \vert} = 2$ we have that $A = \mathbb{Z}_2$ is the unique group of order 2 and the statement holds for $p =2$.

Assume then that the statement has been show for groups of order $\lt n$ and let ${\vert A \vert} = n$.

If $A$ has no non-trivial proper subgroup then $n$ must be prime and $A = \mathbb{Z}_n$ a cyclic group and the statement follows.

If $A$ does have a non-trivial proper subgroup $H \hookrightarrow A$ then $p$ divides either ${\vert H \vert}$ or $\vert A/H\vert$.

In the first case by induction assumption $H$ has an element of order $p$ which is therefore also an element of $G$ of order $p$.

In the second case there is by induction assumption an element $a \in A$ such that $a + H \in A/H$ has order $p$. Since the order of $a + H \in A/H$ divides the order of $a \in A$ it follows that $a$ has order $k p$ for some $k \in \mathbb{N}$. Then $k a$ has order $p$.

Theorem

(fundamental theorem of finite abelian groups)

Every finite abelian group is the direct sum of cyclic groups of prime power order (its p-primary groups).

See for instance (Sullivan).

References

A new proof of the fundamental theorem of finite abelian groups was given in

• Gabriel Navarro, On the fundamental theorem of finite abelian groups, Amer. Math. Monthly, February 2003

reviewed in

• John Sullivan, Classification of finite abelian groups (pdf)