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FiniteAbelianGroupCount[n]

gives the number of finite Abelian groups of order n.

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Scope  
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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:

Wolfram Language code: Table[FiniteAbelianGroupCount[n], {n, 1, 20}]

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

Wolfram Language code: DiscretePlot[FiniteAbelianGroupCount[n], {n, 1, 50}, PlotRange -> {0, 7}]

Scope  (2)

Evaluate for large arguments:

Wolfram Language code: FiniteAbelianGroupCount[2 ^ 500]

FiniteAbelianGroupCount threads element-wise over lists:

Wolfram Language code: FiniteAbelianGroupCount[{2, 4, 6}]

Applications  (2)

Number of non-Abelian groups of order n:

Wolfram Language code: Table[FiniteGroupCount[n] - FiniteAbelianGroupCount[n], {n, 20}]

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

Wolfram Language code: ListPlot[Accumulate[Table[Mod[FiniteAbelianGroupCount[k], 2], {k, 1000}]] / Range[1000]]

Properties & Relations  (6)

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

Wolfram Language code: FiniteAbelianGroupCount[24]

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

Wolfram Language code: FiniteGroupCount[24]

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

Wolfram Language code: FiniteAbelianGroupCount[24]
Wolfram Language code: ab24 = FiniteGroupData[24, "Abelian"]

They all have order 24:

Wolfram Language code: FiniteGroupData[#, "Order"]& /@ ab24

Construct permutation group representations of those groups:

Wolfram Language code: FiniteGroupData[#, "PermutationGroupRepresentation"]& /@ ab24

Check their orders again:

Wolfram Language code: GroupOrder /@ %

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

Wolfram Language code: Times@@PartitionsP[Last /@ FactorInteger[45]]
Wolfram Language code: FiniteAbelianGroupCount[45]

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

Wolfram Language code: FiniteAbelianGroupCount /@ {2 ^ 21, 3 ^ 21, 7 ^ 21, 11 ^ 21}

FiniteAbelianGroupCount is a multiplicative function:

Wolfram Language code: FiniteAbelianGroupCount[3 ^ 2 7 ^ 3]
Wolfram Language code: FiniteAbelianGroupCount[3 ^ 2] FiniteAbelianGroupCount[7 ^ 3]

FindSequenceFunction can recognize the FiniteAbelianGroupCount sequence:

Wolfram Language code: Table[FiniteAbelianGroupCount[n], {n, 10}]
Wolfram Language code: FindSequenceFunction[%, n]

Possible Issues  (1)

FiniteAbelianGroupCount evaluates only for explicit integer values:

Wolfram Language code: FiniteAbelianGroupCount[12 + (E + 1) ^ 2 - Expand[(E + 1) ^ 2]]

Use Simplify to find implicit integers in arguments:

Wolfram Language code: Simplify[%]

Neat Examples  (1)

Successive differences of FiniteAbelianGroupCount modulo 2:

Wolfram Language code: ArrayPlot[Mod[NestList[Differences, FiniteAbelianGroupCount[Range[100]], 100], 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_2026_finiteabeliangroupcount, author="Wolfram Research", title="{FiniteAbelianGroupCount}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_finiteabeliangroupcount, organization={Wolfram Research}, title={FiniteAbelianGroupCount}, year={2008}, url={https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html}, note=[Accessed: 13-June-2026]}