Classification of finite abelian groups

From Groupprops
Revision as of 03:15, 1 January 2012 by Vipul (talk | contribs) (Dependence on partitions of the exponent)
Jump to: navigation, search
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 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.