FiniteAbelianGroupCount

FiniteAbelianGroupCount[n]

gives the number of finite Abelian groups of order n.

Details

  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • FiniteAbelianGroupCount automatically threads over lists.

Examples

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Basic Examples  (2)

Table of values:

Numbers of finite Abelian groups with orders from 1 to 50:

Scope  (2)

Evaluate for large arguments:

FiniteAbelianGroupCount threads element-wise over lists:

Applications  (2)

Number of non-Abelian groups of order n:

Compare cumulative counts of even and odd numbers of Abelian groups:

Properties & Relations  (6)

FiniteAbelianGroupCount[n] gives the number of Abelian groups of order n:

FiniteGroupCount[n] gives the number of groups of order n, both Abelian and non-Abelian:

For low orders, FiniteGroupData lists explicit representative Abelian groups of a given order:

They all have order 24:

Construct permutation group representations of those groups:

Check their orders again:

The number of finite Abelian groups can be found using PartitionsP:

FiniteAbelianGroupCount[n] depends only on prime exponents of n:

FiniteAbelianGroupCount is a multiplicative function:

FindSequenceFunction can recognize the FiniteAbelianGroupCount sequence:

Possible Issues  (1)

FiniteAbelianGroupCount evaluates only for explicit integer values:

Use Simplify to find implicit integers in arguments:

Neat Examples  (1)

Successive differences of FiniteAbelianGroupCount modulo 2:

Wolfram Research (2008), FiniteAbelianGroupCount, Wolfram Language function, https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html.

Text

Wolfram Research (2008), FiniteAbelianGroupCount, Wolfram Language function, https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html.

CMS

Wolfram Language. 2008. "FiniteAbelianGroupCount." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html.

APA

Wolfram Language. (2008). FiniteAbelianGroupCount. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html

BibTeX

@misc{reference.wolfram_2025_finiteabeliangroupcount, author="Wolfram Research", title="{FiniteAbelianGroupCount}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html}", note=[Accessed: 22-February-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_finiteabeliangroupcount, organization={Wolfram Research}, title={FiniteAbelianGroupCount}, year={2008}, url={https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html}, note=[Accessed: 22-February-2025 ]}