# 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 up to 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 prime-base logarithm of order 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 unordered 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.

Thus, the number of abelian groups of order $n$ is given by the product of the numbers: $p(k_1)p(k_2) \dots p(k_t)$

where $p(m)$ denotes the number of unordered integer partitions of $m$.

## Related classifications

connected unipotent abelian algebraic groups of dimension $k$ over algebraic closure of $\mathbb{F}_p$ $\leftrightarrow$ abelian group of prime power order with order $p^k$

The forward direction of the correspondence involves taking the $\mathbb{F}_p$-fixed points of the algebraic group. The number of isomorphism classes on both sides equals the number of unordered integer partitions of $k$, and we also have that:

additive group of truncated ring of Witt vectors to length $m$ $\leftrightarrow$ cyclic group of prime power order $\mathbb{Z}/p^m\mathbb{Z}$