# FiniteAbelianGroupCount

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)

gives the number of Abelian groups of order 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:

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_2024_finiteabeliangroupcount, author="Wolfram Research", title="{FiniteAbelianGroupCount}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html}", note=[Accessed: 21-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_finiteabeliangroupcount, organization={Wolfram Research}, title={FiniteAbelianGroupCount}, year={2008}, url={https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html}, note=[Accessed: 21-June-2024 ]}