Group number function - Revision history

R-a-jones: /* Examples of values */ Added section on squarefree n

2024-08-23T23:47:59Z

Examples of values: Added section on squarefree n

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	* <math>\mathrm{gnu}(2^4)=14</math>, <math>\mathrm{gnu}(p^4)=15</math> for <math>p>2</math>, see [[Classification of groups of prime-fourth order]]		* <math>\mathrm{gnu}(2^4)=14</math>, <math>\mathrm{gnu}(p^4)=15</math> for <math>p>2</math>, see [[Classification of groups of prime-fourth order]]
	* <math>\mathrm{gnu}(2p)=2</math>, see [[classification of groups of order two times a prime]]		* <math>\mathrm{gnu}(2p)=2</math>, see [[classification of groups of order two times a prime]]

			For <math>n</math> a squarefree number, the value of <math>\mathrm{gnu}(n)</math> is given by [[Hölder's formula for the number of groups of squarefree order up to isomorphism\|Hölder's formula]]:

			<math>\mathrm{gnu}(n) = \sum_{m \mid n} \prod_{p \mid n/m} \frac{p^{c(p,m)}-1}{p-1}</math>,

			where <math>p</math> is a prime, and <math>c(p,m)</math> denotes the number of primes <math>q</math> such that <math>q \mid m</math>, <math>q \equiv 1 \mod p</math>.<ref>[https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1121&context=rhumj\| Ganev, Iordan (2010) "Groups of a Square-Free Order,"Rose-Hulman Undergraduate Mathematics Journal: Vol. 11 : Iss. 1 , Article 7.]</ref>

	==Asymptotic bounds==		==Asymptotic bounds==

2024-06-10T17:14:40Z

In mathematical culture

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	==In mathematical culture==		==In mathematical culture==

			{{OEIS\|A000001}}
	The values of the gnu function is the very first sequence in the OEIS, [https://oeis.org/A000001\|A000001].		The values of the gnu function is the very first sequence in the OEIS, [https://oeis.org/A000001\|A000001].

2024-01-12T17:08:49Z

2023-12-09T23:14:21Z

2023-12-09T22:45:53Z

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	See the page [[table of number of groups for small orders]].		See the page [[table of number of groups for small orders]].

			==See also==

			* [[Minimal order attaining function]]

	==References==		==References==

2023-11-21T22:35:56Z

Asymptotic bounds

← Older revision		Revision as of 22:35, 21 November 2023
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	A very weak bound for the number of groups of order <math>n</math> up to isomorphism is <math>\mathrm{gnu}(n) \leq n^{n^2}</math>, because this is simply the number of [[binary operation]]s from a set to itself.		A very weak bound for the number of groups of order <math>n</math> up to isomorphism is <math>\mathrm{gnu}(n) \leq n^{n^2}</math>, because this is simply the number of [[binary operation]]s from a set to itself.

	A better bound that can be proven using elementary methods is <math>\mathrm{gnu}(n) \leq n^{n \log_2 n}</math>. {{~~proof~~\|[[Number of groups of order n up to isomorphism is at most n to the power of (n log base 2 n)]]}}		A better bound that can be proven using elementary methods is <math>\mathrm{gnu}(n) \leq n^{n \log_2 n}</math>. {{proofat\|[[Number of groups of order n up to isomorphism is at most n to the power of (n log base 2 n)]]}}

	===Prime power order===		===Prime power order===

2023-11-21T22:35:37Z

Asymptotic bounds

← Older revision		Revision as of 22:35, 21 November 2023
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	==Asymptotic bounds==		==Asymptotic bounds==

			A very weak bound for the number of groups of order <math>n</math> up to isomorphism is <math>\mathrm{gnu}(n) \leq n^{n^2}</math>, because this is simply the number of [[binary operation]]s from a set to itself.

			A better bound that can be proven using elementary methods is <math>\mathrm{gnu}(n) \leq n^{n \log_2 n}</math>. {{proof\|[[Number of groups of order n up to isomorphism is at most n to the power of (n log base 2 n)]]}}

	===Prime power order===		===Prime power order===

2023-11-20T23:34:24Z

Open problems

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	===Values of the group number function===		===Values of the group number function===

	{{further\|[[~~Category:~~Unclassified group orders]]}}		{{further\|[[Unclassified group orders]]}}

	Certain values of the group number function are unknown, and thus the groups of that order are not classified. The smallest such example is for <math>\mathrm{gnu}(2048)</math>. See [[groups of order 2048]]. We do happen to know that the value of <math>\mathrm{gnu}(2048)</math> strictly exceeds <math>1774274116992170</math>.<ref>[https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf \| John H. Conway, Heiko Dietrich and E.A. O’Brien, Counting groups: gnus, moas and other exotica]</ref>.		Certain values of the group number function are unknown, and thus the groups of that order are not classified. The smallest such example is for <math>\mathrm{gnu}(2048)</math>. See [[groups of order 2048]]. We do happen to know that the value of <math>\mathrm{gnu}(2048)</math> strictly exceeds <math>1774274116992170</math>.<ref>[https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf \| John H. Conway, Heiko Dietrich and E.A. O’Brien, Counting groups: gnus, moas and other exotica]</ref>.

2023-11-20T23:34:13Z

Values of the group number function

← Older revision		Revision as of 23:34, 20 November 2023
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	===Values of the group number function===		===Values of the group number function===

	Certain values of the group number function are unknown, and thus the groups of that order are not classified. The smallest such example is for <math>\mathrm{gnu}(2048)</math>. See [[groups of order 2048]]. We do happen to know that the value of <math>\mathrm{gnu}(2048)</math> strictly exceeds <math>1774274116992170</math>.<ref>[https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf \| John H. Conway, Heiko Dietrich and E.A. O’Brien, Counting groups: gnus, moas and other exotica]</ref>		{{further\|[[Category:Unclassified group orders]]}}

			Certain values of the group number function are unknown, and thus the groups of that order are not classified. The smallest such example is for <math>\mathrm{gnu}(2048)</math>. See [[groups of order 2048]]. We do happen to know that the value of <math>\mathrm{gnu}(2048)</math> strictly exceeds <math>1774274116992170</math>.<ref>[https://www.math.auckland.ac.nz/~obrien/research/gnu.pdf \| John H. Conway, Heiko Dietrich and E.A. O’Brien, Counting groups: gnus, moas and other exotica]</ref>.

	===Non-trivial fixed points of the group number function===		===Non-trivial fixed points of the group number function===

2023-11-10T12:40:37Z

← Older revision		Revision as of 12:40, 10 November 2023
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	The values of the gnu function is the very first sequence in the OEIS, [https://oeis.org/A000001\|A000001].		The values of the gnu function is the very first sequence in the OEIS, [https://oeis.org/A000001\|A000001].

			==Table of values==

			See the page [[table of number of groups for small orders]].

	==References==		==References==