N-abelian group - Revision history

R-a-jones: /* Examples */

2024-12-03T09:37:59Z

Examples

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	===Finite groups===		===Finite groups===

			It follows immediately from [[Lagrange's theorem]] that a [[finite group]] <math>G</math> is <math>\|G\|</math>-abelian. If <math>\|G\|</math> is relatively prime to <math>n(n-1)</math>, then <math>G</math> is <math>n</math>-abelian if and only if it is [[abelian group\|abelian]] ([[N-abelian iff abelian (if order is relatively prime to n(n-1))\|article]])

	We list examples of <math>n</math>-abelian [[finite group]]s for <math>n</math> a positive integer, hence these also give examples of <math>(1-n)</math>-abelian groups by [[n-abelian iff (1-n)-abelian]].		We list examples of <math>n</math>-abelian [[finite group]]s for <math>n</math> a positive integer, hence these also give examples of <math>(1-n)</math>-abelian groups by [[n-abelian iff (1-n)-abelian]].

2024-12-03T09:33:48Z

Finite groups

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	\|-		\|-
	\| 4 \|\| There are 231 4-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest examples are [[dihedral group:D8]] and [[quaternion group]].		\| 4 \|\| There are 231 4-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest examples are [[dihedral group:D8]] and [[quaternion group]].
			\|-
			\| 5 \|\| There are 221 5-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest examples are [[dihedral group:D8]] and [[quaternion group]].
			\|-
			\| 6 \|\| There are 86 6-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest example is [[symmetric group:S3]].
	\|}		\|}

2024-12-03T09:30:09Z

Finite groups

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	\| 2 \|\| no non-abelian finite groups ([[2-abelian iff abelian]])		\| 2 \|\| no non-abelian finite groups ([[2-abelian iff abelian]])
	\|-		\|-
	\| 3 \|\| There are 10 3-abelian non-abelian finite groups with order ~~less than~~ 100 up to isomorphism: [[SmallGroup(27,3)]], [[SmallGroup(27,4)]], [[SmallGroup(54,10)]], [[SmallGroup(54,11)]], [[SmallGroup(81,3)]], [[SmallGroup(81,4)]], [[SmallGroup(81,6)]], [[SmallGroup(81,12)]], [[SmallGroup(81,13)]], [[SmallGroup(81,14)]].		\| 3 \|\| There are 10 3-abelian non-abelian finite groups with order at most 100 up to isomorphism: [[SmallGroup(27,3)]], [[SmallGroup(27,4)]], [[SmallGroup(54,10)]], [[SmallGroup(54,11)]], [[SmallGroup(81,3)]], [[SmallGroup(81,4)]], [[SmallGroup(81,6)]], [[SmallGroup(81,12)]], [[SmallGroup(81,13)]], [[SmallGroup(81,14)]].
			\|-
			\| 4 \|\| There are 231 4-abelian non-abelian finite groups with order at most 100 up to isomorphism. The smallest examples are [[dihedral group:D8]] and [[quaternion group]].
	\|}		\|}

2024-12-03T09:25:49Z

Examples

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	\| 2 \|\| no non-abelian finite groups ([[2-abelian iff abelian]])		\| 2 \|\| no non-abelian finite groups ([[2-abelian iff abelian]])
	\|-		\|-
	\| 3 \|\| There are 10 3-abelian non-abelian finite groups with order less than 100: [[SmallGroup(27,3)]], [[SmallGroup(27,4)]], [[SmallGroup(54,10)]], [[SmallGroup(54,11)]], [[SmallGroup(81,3)]], [[SmallGroup(81,4)]], [[SmallGroup(81,6)]], [[SmallGroup(81,12)]], [[SmallGroup(81,13)]], [[SmallGroup(81,14)]].		\| 3 \|\| There are 10 3-abelian non-abelian finite groups with order less than 100 up to isomorphism: [[SmallGroup(27,3)]], [[SmallGroup(27,4)]], [[SmallGroup(54,10)]], [[SmallGroup(54,11)]], [[SmallGroup(81,3)]], [[SmallGroup(81,4)]], [[SmallGroup(81,6)]], [[SmallGroup(81,12)]], [[SmallGroup(81,13)]], [[SmallGroup(81,14)]].
	\|}		\|}

2024-12-03T09:25:09Z

Finite groups: fixed table

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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! <math>n</math>!! Non-abelian <math>n</math> abelian groups. !!		! <math>n</math>!! Non-abelian <math>n</math>-abelian groups.
	\|-		\|-
	\| 1 \|\| all non-abelian finite groups		\| 1 \|\| all non-abelian finite groups

2024-12-03T09:24:47Z

some examples

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	* [[n-nilpotent group]]		* [[n-nilpotent group]]
	* [[n-solvable group]]		* [[n-solvable group]]

			==Examples==

			===Finite groups===

			We list examples of <math>n</math>-abelian [[finite group]]s for <math>n</math> a positive integer, hence these also give examples of <math>(1-n)</math>-abelian groups by [[n-abelian iff (1-n)-abelian]].

			We list non-abelian examples of finite groups here only, all the abelian finite groups are trivially <math>n</math>-abelian for any given <math>n</math>.

			{\| class="sortable" border="1"
			! <math>n</math>!! Non-abelian <math>n</math> abelian groups. !!
			\|-
			\| 1 \|\| all non-abelian finite groups
			\|-
			\| 2 \|\| no non-abelian finite groups ([[2-abelian iff abelian]])
			\|-
			\| 3 \|\| There are 10 3-abelian non-abelian finite groups with order less than 100: [[SmallGroup(27,3)]], [[SmallGroup(27,4)]], [[SmallGroup(54,10)]], [[SmallGroup(54,11)]], [[SmallGroup(81,3)]], [[SmallGroup(81,4)]], [[SmallGroup(81,6)]], [[SmallGroup(81,12)]], [[SmallGroup(81,13)]], [[SmallGroup(81,14)]].
			\|}

2012-08-10T21:05:43Z

Definition

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	Suppose <math>n</math> is an integer. A [[group]] <math>G</math> is termed a '''<math>n</math>-abelian group''' if the <math>n^{th}</math> power map <math>x \mapsto x^n</math> is an [[endomorphism]] of <math>G</math>, i.e., <math>(xy)^n = x^ny^n</math> for all <math>x,y \in G</math>. If this is the case, then the <math>n^{th}</math> power map is termed a [[universal power endomorphism]] of <math>G</math>.		Suppose <math>n</math> is an integer. A [[group]] <math>G</math> is termed a '''<math>n</math>-abelian group''' if the <math>n^{th}</math> power map <math>x \mapsto x^n</math> is an [[endomorphism]] of <math>G</math>, i.e., <math>(xy)^n = x^ny^n</math> for all <math>x,y \in G</math>. If this is the case, then the <math>n^{th}</math> power map is termed a [[universal power endomorphism]] of <math>G</math>.

			As noted below, [[n-abelian iff (1-n)-abelian]], so it suffices to restrict attention to <math>n</math> a positive integer.
	===Alternative definitions===		===Alternative definitions===

2012-08-10T21:05:02Z

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			{{natural number-parametrized group property}}
	==Definition==		==Definition==

2012-08-10T21:04:39Z

Definition

← Older revision		Revision as of 21:04, 10 August 2012
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	===Alternative definitions===		===Alternative definitions===

	See [[Alperin's structure ~~theorems~~ for n-abelian groups]].		See [[Alperin's structure theorem for n-abelian groups]].

	==Facts==		==Facts==

2012-08-10T20:57:08Z

Particular values

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	\| -1 \|\| [[abelian group]]s only \|\| [[-1-abelian iff abelian]] \|\|		\| -1 \|\| [[abelian group]]s only \|\| [[-1-abelian iff abelian]] \|\|
	\|-		\|-
	\| 3 \|\| [[2-Engel group]] and [[derived subgroup]] has exponent dividing three \|\| [[Levi's characterization of 3-abelian groups]] \|\| [[cube map is surjective endomorphism implies abelian]], [[cube map is endomorphism iff abelian (if order is not a multiple of 3)]]		\| 3 \|\| [[3-abelian group]] means: [[2-Engel group]] and [[derived subgroup]] has exponent dividing three \|\| [[Levi's characterization of 3-abelian groups]] \|\| [[cube map is surjective endomorphism implies abelian]], [[cube map is endomorphism iff abelian (if order is not a multiple of 3)]], [[cube map is endomorphism implies class three]]
	\|-		\|-
	\| -2 \|\| same as for 3-abelian \|\| (based on [[n-abelian iff (1-n)-abelian]]) \|\|		\| -2 \|\| same as for 3-abelian \|\| (based on [[n-abelian iff (1-n)-abelian]]) \|\|