Classification of finite abelian groups

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

Goal

Our goal in this article is to give a complete description of all finite abelian groups. This includes:

• 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

• Classification of connected unipotent abelian algebraic groups over an algebraically closed field: The prime characteristic case of this is essentially the same as the classification of finite abelian groups. Specifically, there is a correspondence:

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 $\mathbb{Z}/p^m\mathbb{Z}$