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finite abelian group

Definition

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

Properties

Proposition

If a finite abelian group $A$ has order $\mid A\mid =p$ 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 $\mid A\mid$, 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 $\mid A\mid =2$ 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 $<n$ and let $\mid A\mid =n$.

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 $\mid H\mid$ or $\mid A/H\mid$.

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 $kp$ for some $k\in \mathbb{N}$. Then $ka$ has order $p$.

Theorem

Every finite abelian group is the direct sum of cyclic groups of order $p^k$ for a prime number $p\in \mathbb{N}$.

See for instance (Sullivan).

References

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