finite abelian group




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



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


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

This is Cauchy's theorem restricted to abelian groups.


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

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

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

If AA does have a non-trivial proper subgroup HAH \hookrightarrow A then pp divides either |H|{\vert H \vert} or |A/H|\vert A/H\vert.

In the first case by induction assumption HH has an element of order pp which is therefore also an element of GG of order pp.

In the second case there is by induction assumption an element aAa \in A such that a+HA/Ha + H \in A/H has order pp. Since the order of a+HA/Ha + H \in A/H divides the order of aAa \in A it follows that aa has order kpk p for some kk \in \mathbb{N}. Then kak a has order pp.


(fundamental theorem of finite abelian groups)

Every finite abelian group is the direct sum of cyclic groups of order p kp^k for a prime number pp \in \mathbb{N} (its p-primary groups).

See for instance (Sullivan).


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

reviewed in

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

Last revised on July 13, 2016 at 14:45:25. See the history of this page for a list of all contributions to it.