Minimal order attaining function - Revision history

R-a-jones: /* Small values */

2024-05-26T17:32:09Z

Small values

← Older revision		Revision as of 17:32, 26 May 2024
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	This function is not well understood, and some very small values of the function are not known. For example, it is conjectured that <math>\mathrm{moa}(33)=163293</math>. This is the smallest such unknown value. As of ~~December 2023~~, this has not been confirmed.		This function is not well understood, and some very small values of the function are not known. For example, it is conjectured that <math>\mathrm{moa}(33)=163293</math>. This is the smallest such unknown value. As of May 2024, this has not been confirmed.

	==See also==		==See also==

2024-01-12T17:08:59Z

2023-12-13T12:58:26Z

Question of well-definedness

← Older revision		Revision as of 12:58, 13 December 2023
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	{{see also\|[[Group number function #Open problems]]}}		{{see also\|[[Group number function #Open problems]]}}

	It is not known whether <math>\mathrm{moa}</math> is well-defined for all natural numbers. Equivalently, it is a conjecture as to whether or not the group number functions is surjective. In other words, it is possible that there is such a number for which there will never be that number of groups up to isomorphism of ~~a fixed~~ order.		It is not known whether <math>\mathrm{moa}</math> is well-defined for all natural numbers. Equivalently, it is a conjecture as to whether or not the group number functions is surjective. In other words, it is possible that there is such a number for which there will never be that number of groups up to isomorphism of any given order.

	==Small values==		==Small values==

2023-12-09T23:14:08Z

2023-12-09T23:12:26Z

Question of well-definedness

← Older revision		Revision as of 23:12, 9 December 2023
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	{{see also\|[[Group number function #Open problems]]}}		{{see also\|[[Group number function #Open problems]]}}

	It is not known whether <math>\mathrm{moa}</math> is well-defined for all natural numbers - it is a conjecture as to whether or not the group number functions is surjective. In other words, it is possible that there is such a number for which there will never be that number of groups up to isomorphism of a fixed order.		It is not known whether <math>\mathrm{moa}</math> is well-defined for all natural numbers. Equivalently, it is a conjecture as to whether or not the group number functions is surjective. In other words, it is possible that there is such a number for which there will never be that number of groups up to isomorphism of a fixed order.

	==Small values==		==Small values==

2023-12-09T23:12:10Z

In terms of the group number function

← Older revision		Revision as of 23:12, 9 December 2023
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	===In terms of the group number function===		===In terms of the group number function===

	We can define the minimal order attaining function in terms of the [[group number function]], denoted \mathrm{gnu}, which outputs the number of groups of a given order up to isomorphism:		We can define the minimal order attaining function in terms of the [[group number function]], denoted <math>\mathrm{gnu}</math>, which outputs the number of groups of a given order up to isomorphism:

	<math>\mathrm{moa}(n) = \mathrm{min} \{ m \in \N: \mathrm{gnu}(m)=n \}</math>.		<math>\mathrm{moa}(n) = \mathrm{min} \{ m \in \N: \mathrm{gnu}(m)=n \}</math>.

2023-12-09T23:08:26Z

Small values

← Older revision		Revision as of 23:08, 9 December 2023
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	\| 10 \|\| 90 \|\| [[groups of order 90]]		\| 10 \|\| 90 \|\| [[groups of order 90]]
	\|}		\|}

			This function is not well understood, and some very small values of the function are not known. For example, it is conjectured that <math>\mathrm{moa}(33)=163293</math>. This is the smallest such unknown value. As of December 2023, this has not been confirmed.

	==See also==		==See also==

2023-12-09T23:04:02Z

Small values

← Older revision		Revision as of 23:04, 9 December 2023
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	\| 10 \|\| 90 \|\| [[groups of order 90]]		\| 10 \|\| 90 \|\| [[groups of order 90]]
	\|}		\|}

			==See also==

			* [[Group number function]]
			==External links==

			The sequence [[Oeis:A046057]] in the Online Encyclopedia of Integer Sequences gives the values of the minimal order attaining function.

2023-12-09T22:53:35Z

Question of well-definedness

← Older revision		Revision as of 22:53, 9 December 2023
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	===Question of well-definedness===		===Question of well-definedness===

	{{see also\|[[Group number function # Open problems]]}}		{{see also\|[[Group number function #Open problems]]}}

	It is not known whether <math>\mathrm{moa}</math> is well-defined for all natural numbers - it is a conjecture as to whether or not the group number functions is surjective. In other words, it is possible that there is such a number for which there will never be that number of groups up to isomorphism of a fixed order.		It is not known whether <math>\mathrm{moa}</math> is well-defined for all natural numbers - it is a conjecture as to whether or not the group number functions is surjective. In other words, it is possible that there is such a number for which there will never be that number of groups up to isomorphism of a fixed order.

2023-12-09T22:53:22Z

Question of well-definedness

← Older revision		Revision as of 22:53, 9 December 2023
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	===Question of well-definedness===		===Question of well-definedness===

			{{see also\|[[Group number function # Open problems]]}}

	It is not known whether <math>\mathrm{moa}</math> is well-defined for all natural numbers - it is a conjecture as to whether or not the group number functions is surjective. In other words, it is possible that there is such a number for which there will never be that number of groups up to isomorphism of a fixed order.		It is not known whether <math>\mathrm{moa}</math> is well-defined for all natural numbers - it is a conjecture as to whether or not the group number functions is surjective. In other words, it is possible that there is such a number for which there will never be that number of groups up to isomorphism of a fixed order.