# Classification of finite abelian groups

This is a survey article related to:Abelianness
View other survey articles about Abelianness

## Goal

• Describing each finite Abelian group in an easy way from which all questions about its structure can be answered
• For every natural number, giving a complete list of all the isomorphism classes of Abelian groups having that natural number as order.

## Structure theorem

Further information: structure theorem for finitely generated Abelian groups

This theorem is the main result that gives the complete classification. We state it here in a form that is suited for the classification:

Every finite Abelian group can be expressed as a product of cyclic groups of prime power order. Moreover this expression is unique upto ordering of the factors and upto isomorphism

## Classification

### Reduction to case of prime power order groups

The above theorem also tells us that a finite Abelian group is expressible as a direct product of its Sylow subgroups, so it suffices for us to classify all Abelian groups of prime power order.

### Dependence on partitions of the exponent

If an Abelian group of prime power order $p^k$ is expressed as a direct product of cyclic groups of prime power order then the sum of the exponents of all the direct factors equals $k$. Conversely, given any partition of $k$ into nonnegative integers, say $k = m_1 + m_2 + \ldots + m_r$, we can form an Abelian group:

$\Z/p^{m_1}\Z \times \Z/p^{m_2}\Z \times \ldots \Z/p^{m_r}\Z$

Thus the set of Abelian groups of order $p^k$ is in bijection with the set of nonnegative integer partitions of $k$.

### The overall description

Let $n = p_1^{k_1}p_2^{k_2} \ldots p_t^{k_t}$

Then the set of Abelian groups of order $n$ is in bijective correspondence with $P(k_1) \times P(k_2) \times \ldots \times P(k_t)$ where $P(m)$ denotes the set of unordered integer partitions of the integer $m$ into nonnegative integer parts.