Sylow does not satisfy transfer condition - Revision history

Vipul: /* Property-theoretic proof */

2009-02-14T23:36:42Z

Property-theoretic proof

← Older revision		Revision as of 23:36, 14 February 2009
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	===Property-theoretic proof===		===Property-theoretic proof===

	We know that the property of being a Sylow subgroup is transitive (a Sylow subgroup of a Sylow subgroup is Sylow). Thus, if the property of being Sylow satisfies the transfer condition, we have that the property of being a Sylow subgroup is intersection-closed, by the general fact [[Transitive and transfer condition implies intersection-closed]].		We know that the property of being a Sylow subgroup is transitive (a Sylow subgroup of a Sylow subgroup is Sylow). Thus, if the property of being Sylow satisfies the transfer condition, we have that the property of being a Sylow subgroup is intersection-closed, by the general fact [[Transitive and transfer condition implies finite-intersection-closed]].

	On the other hand, an intersection of Sylow subgroups need not be Sylow.		On the other hand, an intersection of Sylow subgroups need not be Sylow.

Vipul at 13:48, 28 September 2008

2008-09-28T13:48:34Z

← Older revision		Revision as of 13:48, 28 September 2008
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	* [[Hall does not satisfy transfer condition]]		* [[Hall does not satisfy transfer condition]]
			* [[Normal Sylow satisfies transfer condition]]
			* [[Normal Hall satisfies transfer condition]]
	==Proof==		==Proof==

Vipul at 21:43, 19 September 2008

2008-09-19T21:43:47Z

← Older revision		Revision as of 21:43, 19 September 2008
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	{{subgroup metaproperty dissatisfaction in\|		{{subgroup metaproperty dissatisfaction\|
	group property = finite group\|		group property = finite group\|
	property = Sylow subgroup\|		property = Sylow subgroup\|
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	It is possible to have a [[finite group]] <math>G</math>, a Sylow subgroup <math>H</math>, and a subgroup <math>K</math> of <math>G</math> such that <math>H \cap K</math> is not a Sylow subgroup of <math>K</math>.		It is possible to have a [[finite group]] <math>G</math>, a Sylow subgroup <math>H</math>, and a subgroup <math>K</math> of <math>G</math> such that <math>H \cap K</math> is not a Sylow subgroup of <math>K</math>.

			==Related facts==

			* [[Hall does not satisfy transfer condition]]

	==Proof==		==Proof==

Vipul: New page: {{subgroup metaproperty dissatisfaction in| group property = finite group| property = Sylow subgroup| metaproperty = transfer condition}} ==Statement== It is possible to have a [[finite ...

2008-09-19T21:42:43Z

New page: {{subgroup metaproperty dissatisfaction in| group property = finite group| property = Sylow subgroup| metaproperty = transfer condition}} ==Statement== It is possible to have a [[finite ...

New page

{{subgroup metaproperty dissatisfaction in|
group property = finite group|
property = Sylow subgroup|
metaproperty = transfer condition}}

==Statement==

It is possible to have a [[finite group]] <math>G</math>, a Sylow subgroup <math>H</math>, and a subgroup <math>K</math> of <math>G</math> such that <math>H \cap K</math> is not a Sylow subgroup of <math>K</math>.

==Proof==

===Hands-on proof===

===Property-theoretic proof===

We know that the property of being a Sylow subgroup is transitive (a Sylow subgroup of a Sylow subgroup is Sylow). Thus, if the property of being Sylow satisfies the transfer condition, we have that the property of being a Sylow subgroup is intersection-closed, by the general fact [[Transitive and transfer condition implies intersection-closed]].

On the other hand, an intersection of Sylow subgroups need not be Sylow.