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FiniteAbelianGroupCount
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FiniteAbelianGroupCount

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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

open allclose all

Basic Examples  (2)Summary of the most common use cases

Table of values:

In[1]:=1
✖
https://wolfram.com/xid/0vu4cy96rx6svadz0ya-f0k9f7
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-39bhoi
Out[1]=1

Scope  (2)Survey of the scope of standard use cases

Evaluate for large arguments:

In[1]:=1
✖
https://wolfram.com/xid/0vu4cy96rx6svadz0ya-9ogmq
Out[1]=1

FiniteAbelianGroupCount threads element-wise over lists:

In[1]:=1
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-hx7zo
Out[1]=1

Applications  (2)Sample problems that can be solved with this function

Number of non-Abelian groups of order n:

In[1]:=1
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-em2hot
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-77omz
Out[1]=1

Properties & Relations  (6)Properties of the function, and connections to other functions

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

In[1]:=1
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-059mf3
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-fed55
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-opip95
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-xeu24p
Out[2]=2

They all have order 24:

In[3]:=3
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-nhotvj
Out[3]=3

Construct permutation group representations of those groups:

In[4]:=4
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-6ql8kl
Out[4]=4

Check their orders again:

In[5]:=5
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-see2tr
Out[5]=5

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

In[1]:=1
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-gsvedy
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-p127ee
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-bpsz7j
Out[1]=1

FiniteAbelianGroupCount is a multiplicative function:

In[1]:=1
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-c3hk8p
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-hmrxv1
Out[2]=2

FindSequenceFunction can recognize the FiniteAbelianGroupCount sequence:

In[1]:=1
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-hj2mn6
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-5okec
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

FiniteAbelianGroupCount evaluates only for explicit integer values:

In[1]:=1
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-bft5sy
Out[1]=1

Use Simplify to find implicit integers in arguments:

In[2]:=2
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https://wolfram.com/xid/0vu4cy96rx6svadz0ya-h7r3kf
Out[2]=2

Neat Examples  (1)Surprising or curious use cases

Successive differences of FiniteAbelianGroupCount modulo 2:

In[1]:=1
✖
https://wolfram.com/xid/0vu4cy96rx6svadz0ya-mfji4
Out[1]=1
Wolfram Research (2008), FiniteAbelianGroupCount, Wolfram Language function, https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html.
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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.

✖
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.

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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

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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: 07-June-2025 ]}

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@misc{reference.wolfram_2025_finiteabeliangroupcount, author="Wolfram Research", title="{FiniteAbelianGroupCount}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html}", note=[Accessed: 07-June-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: 07-June-2025 ]}

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@online{reference.wolfram_2025_finiteabeliangroupcount, organization={Wolfram Research}, title={FiniteAbelianGroupCount}, year={2008}, url={https://reference.wolfram.com/language/ref/FiniteAbelianGroupCount.html}, note=[Accessed: 07-June-2025 ]}