Inverse map is automorphism iff abelian - Revision history

Vipul: /* Similar facts for other power maps */

2012-08-10T20:13:57Z

Similar facts for other power maps

← Older revision		Revision as of 20:13, 10 August 2012
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	===Similar facts for other power maps===		===Similar facts for other power maps===

	* [[Square map is ~~endomorphism iff abelian]]~~		We say that a group is a [[n-abelian group]] if the <math>n^{th}</math> power map is an endomorphism. Here are some related facts about <math>n</math>-abelian groups.
	* [[Cube map is endomorphism iff abelian (if order is not a ~~multiple of 3)]],~~ [[~~cube map is automorphism implies~~ abelian]]
	* [[kth power map is bijective iff k is relatively prime to the ~~order]]~~		{{#lst:n-abelian group\|general facts}}
	* [[nth power map is endomorphism ~~iff abelian (if order is relatively prime to n(~~n-~~1))]]~~		{{#lst:n-abelian group\|particular values}}
	* [[abelian implies universal power map is endomorphism]]
	* [[Frattini-in-center odd-order p-group ~~implies (p plus 1)-power map is automorphism]]~~
	* [[Frattini-in-center odd-order p-group ~~implies p-power map is endomorphism]]~~

	===Applications===		===Applications===

2011-02-24T18:40:28Z

Related facts for other algebraic structures

← Older revision		Revision as of 18:40, 24 February 2011
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	===Related facts for other algebraic structures===		===Related facts for other algebraic structures===

	* [[Multiplication by n map is endomorphism iff derived subring has exponent dividing n(n-1)]] for [[Lie ring]]s. In particular, we can construct examples of non-abelian Lie rings where the negation map is an automorphism.		* [[Multiplication by n map is an endomorphism iff derived subring has exponent dividing n(n-1)]] for [[Lie ring]]s. In particular, we can construct examples of non-abelian Lie rings where the negation map is an automorphism.
	* [[Inverse map is automorphism not implies abelian for loop]]		* [[Inverse map is automorphism not implies abelian for loop]]

2011-02-24T18:39:59Z

Related facts for other algebraic structures

← Older revision		Revision as of 18:39, 24 February 2011
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	===Related facts for other algebraic structures===		===Related facts for other algebraic structures===

	* [[~~Negation~~ map is ~~automorphism~~ iff ~~abelian~~ for Lie ring]]		* [[Multiplication by n map is endomorphism iff derived subring has exponent dividing n(n-1)]] for [[Lie ring]]s. In particular, we can construct examples of non-abelian Lie rings where the negation map is an automorphism.
	* [[Inverse map is automorphism not implies abelian for loop]]		* [[Inverse map is automorphism not implies abelian for loop]]

2011-02-24T18:36:19Z

Related facts

← Older revision		Revision as of 18:36, 24 February 2011
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	* [[Square map is endomorphism iff abelian]]		* [[Square map is endomorphism iff abelian]]
	* [[Cube map is endomorphism iff abelian (if order is not a multiple of 3)]]		* [[Cube map is endomorphism iff abelian (if order is not a multiple of 3)]], [[cube map is automorphism implies abelian]]
			* [[kth power map is bijective iff k is relatively prime to the order]]
			* [[nth power map is endomorphism iff abelian (if order is relatively prime to n(n-1))]]
			* [[abelian implies universal power map is endomorphism]]
			* [[Frattini-in-center odd-order p-group implies (p plus 1)-power map is automorphism]]
			* [[Frattini-in-center odd-order p-group implies p-power map is endomorphism]]

			===Applications===

	* [[Fixed-point-free involution on finite group is inverse map]]		* [[Fixed-point-free involution on finite group is inverse map]]
	* [[Automorphism sends more than three-fourths of elements to inverses implies abelian]]		* [[Automorphism sends more than three-fourths of elements to inverses implies abelian]]

2011-02-24T18:32:52Z

Similar facts for other algebraic structures

← Older revision		Revision as of 18:32, 24 February 2011
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	* [[Automorphism sends more than three-fourths of elements to inverses implies abelian]]		* [[Automorphism sends more than three-fourths of elements to inverses implies abelian]]

	===~~Similar~~ facts for other algebraic structures===		===Related facts for other algebraic structures===

	* [[Negation map is automorphism iff abelian for Lie ring]]		* [[Negation map is automorphism iff abelian for Lie ring]]

2011-02-24T18:32:40Z

Related facts

← Older revision		Revision as of 18:32, 24 February 2011
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	==Related facts==		==Related facts==

			===Similar facts for other power maps===

	* [[Square map is endomorphism iff abelian]]		* [[Square map is endomorphism iff abelian]]
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	* [[Fixed-point-free involution on finite group is inverse map]]		* [[Fixed-point-free involution on finite group is inverse map]]
	* [[Automorphism sends more than three-fourths of elements to inverses implies abelian]]		* [[Automorphism sends more than three-fourths of elements to inverses implies abelian]]

			===Similar facts for other algebraic structures===

			* [[Negation map is automorphism iff abelian for Lie ring]]
			* [[Inverse map is automorphism not implies abelian for loop]]

	==Proof==		==Proof==

2011-02-24T16:55:32Z

← Older revision		Revision as of 16:55, 24 February 2011
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	{{elementary nonbasic fact}}		{{elementary nonbasic fact}}
	{{automorphism group control result}}		{{automorphism group control result}}
			[[difficulty level::1\| ]]
	==Statement==		==Statement==

2010-09-26T15:29:55Z

Related facts

← Older revision		Revision as of 15:29, 26 September 2010
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	* [[Cube map is endomorphism iff abelian (if order is not a multiple of 3)]]		* [[Cube map is endomorphism iff abelian (if order is not a multiple of 3)]]
	* [[Fixed-point-free involution on finite group is inverse map]]		* [[Fixed-point-free involution on finite group is inverse map]]
			* [[Automorphism sends more than three-fourths of elements to inverses implies abelian]]

	==Proof==		==Proof==

2009-01-21T23:15:17Z

← Older revision		Revision as of 23:15, 21 January 2009
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	# The map sending every element to its inverse, is an [[endomorphism]]		# The map sending every element to its inverse, is an [[endomorphism]]
	# The map sending every element to its inverse, is an [[automorphism]]		# The map sending every element to its inverse, is an [[automorphism]]
	# The group is [[~~Abelian~~ group\|~~Abelian~~]]		# The group is [[abelian group\|abelian]]

	The equivalence of (1) and (2) is direct from the fact that the inverse map is bijective.		The equivalence of (1) and (2) is direct from the fact that the inverse map is bijective.
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	==Related facts==		==Related facts==

	* [[Square map is endomorphism iff ~~Abelian~~]]		* [[Square map is endomorphism iff abelian]]
	* [[Cube map is endomorphism iff ~~Abelian~~ (if order is not a multiple of 3)]]		* [[Cube map is endomorphism iff abelian (if order is not a multiple of 3)]]
	* [[Fixed-point-free involution on finite group is inverse map]]		* [[Fixed-point-free involution on finite group is inverse map]]

2009-01-21T23:06:53Z

← Older revision	Revision as of 23:06, 21 January 2009
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