Hall subgroup - Revision history

Vipul: /* Weaker properties */

2014-12-21T18:16:48Z

Weaker properties

← Older revision		Revision as of 18:16, 21 December 2014
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	\| [[Stronger than::closure-characteristic subgroup]] \|\| [[normal closure]] is a [[characteristic subgroup]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Hall subgroup}}		\| [[Stronger than::closure-characteristic subgroup]] \|\| [[normal closure]] is a [[characteristic subgroup]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Hall subgroup}}
	\|-		\|-
	\| [[Stronger than::paracharacteristic subgroup]] \|\| \|\| [[Hall implies paracharacteristic]] \|\| \|\| {{intermediate notions short\|paracharacteristic subgroup\|Hall subgroup}}		\| [[Stronger than::paracharacteristic subgroup]] \|\| [[contranormal subgroup\|contranormal]] in its join with any [[automorphic subgroups\|automorphic subgroup]] \|\| (via join of Sylow subgroups, see also [[Hall implies paracharacteristic]]) \|\| \|\| {{intermediate notions short\|paracharacteristic subgroup\|Hall subgroup}}
	\|-		\|-
	\| [[Stronger than::paranormal subgroup]] \|\| \|\| (via paracharacteristic) \|\| \|\| {{intermediate notions short\|paranormal subgroup\|Hall subgroup}}		\| [[Stronger than::paranormal subgroup]] \|\| [[contranormal subgroup\|contranormal]] in its join with any [[conjugate subgroups\|conjugate subgroup]] \|\| (via paracharacteristic) \|\| \|\| {{intermediate notions short\|paranormal subgroup\|Hall subgroup}}
	\|-		\|-
	\| [[Stronger than::polycharacteristic subgroup]] \|\| \|\| (via paracharacteristic) \|\| \|\| {{intermediate notions short\|polycharacteristic subgroup\|Hall subgroup}}		\| [[Stronger than::polycharacteristic subgroup]] \|\| \|\| (via paracharacteristic) \|\| \|\| {{intermediate notions short\|polycharacteristic subgroup\|Hall subgroup}}

Vipul: /* Weaker properties */

2014-12-21T17:48:23Z

Weaker properties

← Older revision		Revision as of 17:48, 21 December 2014
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	\| [[Stronger than::paracharacteristic subgroup]] \|\| \|\| [[Hall implies paracharacteristic]] \|\| \|\| {{intermediate notions short\|paracharacteristic subgroup\|Hall subgroup}}		\| [[Stronger than::paracharacteristic subgroup]] \|\| \|\| [[Hall implies paracharacteristic]] \|\| \|\| {{intermediate notions short\|paracharacteristic subgroup\|Hall subgroup}}
	\|-		\|-
	\| [[Stronger than::paranormal subgroup]] \|\| \|\| (via paracharacteristic~~, see also [[Hall implies paranormal]]~~) \|\| \|\| {{intermediate notions short\|paranormal subgroup\|Hall subgroup}}		\| [[Stronger than::paranormal subgroup]] \|\| \|\| (via paracharacteristic) \|\| \|\| {{intermediate notions short\|paranormal subgroup\|Hall subgroup}}
	\|-		\|-
	\| [[Stronger than::polycharacteristic subgroup]] \|\| \|\| (via paracharacteristic) \|\| \|\| {{intermediate notions short\|polycharacteristic subgroup\|Hall subgroup}}		\| [[Stronger than::polycharacteristic subgroup]] \|\| \|\| (via paracharacteristic) \|\| \|\| {{intermediate notions short\|polycharacteristic subgroup\|Hall subgroup}}

Vipul: /* Weaker properties */

2014-12-21T17:47:47Z

Weaker properties

← Older revision		Revision as of 17:47, 21 December 2014
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	===Weaker properties===		===Weaker properties===

	* [[Stronger than::~~Join~~ of Sylow subgroups]]~~: {{proofat~~\|[[Hall implies join of Sylow subgroups]]}}		{\| class="sortable" border="1"
	* [[Stronger than::~~Join~~ of automorph-conjugate subgroups]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Stronger than::~~Core~~-characteristic subgroup]]		\|-
	* [[Stronger than::~~Closure~~-characteristic subgroup]]		\| [[Stronger than::join of Sylow subgroups]] \|\| join of [[Sylow subgroup]]s of the whole group \|\| [[Hall implies join of Sylow subgroups]] \|\| \|\| {{intermediate notions short\|join of Sylow subgroups\|Hall subgroup}}
	* [[Stronger than::~~Paracharacteristic~~ subgroup]]~~: {{proofat~~\|[[Hall implies paracharacteristic]]}}		\|-
	* [[Stronger than::~~Paranormal~~ subgroup]]~~: {{proofat~~\|[[Hall implies paranormal]]}}		\| [[Stronger than::join of automorph-conjugate subgroups]] \|\| join of [[automorph-conjugate subgroup]]s of the whole group \|\| (via join of Sylow sugroups) \|\| \|\| {{intermediate notions short\|join of automorph-conjugate subgroups\|Hall subgroup}}
	* [[Stronger than::~~Polycharacteristic~~ subgroup]]		\|-
	* [[Stronger than::~~Polynormal~~ subgroup]]		\| [[Stronger than::core-characteristic subgroup]] \|\| [[normal core]] is a [[characteristic subgroup]] \|\| \|\| \|\| {{intermediate notions short\|core-characteristic subgroup\|Hall subgroup}}
	* [[Stronger than::~~Intermediately~~ normal-to-characteristic subgroup]]		\|-
	* [[Stronger than::~~Intermediately~~ subnormal-to-normal subgroup]]		\| [[Stronger than::closure-characteristic subgroup]] \|\| [[normal closure]] is a [[characteristic subgroup]] \|\| \|\| \|\| {{intermediate notions short\|closure-characteristic subgroup\|Hall subgroup}}
			\|-
			\| [[Stronger than::paracharacteristic subgroup]] \|\| \|\| [[Hall implies paracharacteristic]] \|\| \|\| {{intermediate notions short\|paracharacteristic subgroup\|Hall subgroup}}
			\|-
			\| [[Stronger than::paranormal subgroup]] \|\| \|\| (via paracharacteristic, see also [[Hall implies paranormal]]) \|\| \|\| {{intermediate notions short\|paranormal subgroup\|Hall subgroup}}
			\|-
			\| [[Stronger than::polycharacteristic subgroup]] \|\| \|\| (via paracharacteristic) \|\| \|\| {{intermediate notions short\|polycharacteristic subgroup\|Hall subgroup}}
			\|-
			\| [[Stronger than::polynormal subgroup]] \|\| \|\| (via paranoral, also via polycharacteristic) \|\| \|\| {{intermediate notions short\|polynormal subgroup\|Hall subgroup}}
			\|-
			\| [[Stronger than::intermediately normal-to-characteristic subgroup]] \|\| for any intermediate subgroup in which it is [[normal subgroup\|normal]], it is also [[characteristic subgroup\|characteristic]] in that subgroup \|\| \|\| \|\| {{intermediate notions short\|intermediately normal-to-characteristic subgroup\|Hall subgroup}}
			\|-
			\| [[Stronger than::intermediately subnormal-to-normal subgroup]] \|\| for any intermediate subgroup in which it is [[subnormal subgroup\|subnormal]], it is also [[normal subgroup\|normal]] in that subgroup \|\| \|\| \|\| {{intermediate notions short\|intermediately subnormal-to-normal subgroup\|Hall subgroup}}
			\|}

	===Conjunction with other properties===		===Conjunction with other properties===

Xubia at 13:53, 11 September 2014

2014-09-11T13:53:37Z

← Older revision		Revision as of 13:53, 11 September 2014
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			{{semibasicdef}}
			{{subgroup property}}
			[[importance rank::2\| ]]

	==Definition==		==Definition==

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	{{#ask: [[satisfies property::Hall subgroup]][[Category:Particular subgroups]]\|?group part\|?subgroup part\|?quotient part}}		{{#ask: [[satisfies property::Hall subgroup]][[Category:Particular subgroups]]\|?group part\|?subgroup part\|?quotient part}}

	~~{{semibasicdef}}~~
	~~{{subgroup property}}~~
	~~[[importance rank::2\| ]]~~

	==Facts==		==Facts==

Xubia: /* Existence and domination */

2014-09-11T13:49:49Z

Existence and domination

← Older revision		Revision as of 13:49, 11 September 2014
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	===Existence and domination===		===Existence and domination===

	* [[Existence of pi-subgroups for all prime sets pi is equivalent to existence of p-complements for all primes p]]		* [[Existence of pi-subgroups for all prime sets <math>\pi</math> is equivalent to existence of p-complements for all primes p]]
	* [[ECD condition for pi-subgroups in finite solvable groups]]: This states that in [[finite solvable group]]s, <math>\pi</math>-Hall subgroup exist for all prime sets <math>\pi</math>, they are conjugate, and they dominate <math>\pi</matH>-subgroups.		* [[ECD condition for <math>\pi</math>-subgroups in finite solvable groups]]: This states that in [[finite solvable group]]s, <math>\pi</math>-Hall subgroup exist for all prime sets <math>\pi</math>, they are conjugate, and they dominate <math>\pi</matH>-subgroups.
	* [[Hall's theorem]]: This is a converse to the above, stating that if <math>\pi</math>-Hall subgroups exist for all prime sets <math>\pi</math>, then the group is solvable.		* [[Hall's theorem]]: This is a converse to the above, stating that if <math>\pi</math>-Hall subgroups exist for all prime sets <math>\pi</math>, then the group is solvable.

Vipul: /* Particular examples */

2012-03-23T00:45:09Z

Particular examples

← Older revision		Revision as of 00:45, 23 March 2012
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	* [[A4 in A5]]: The subgroup has order 12 and index 5 in a group of order 60. It is a <math>\{ 2,3 \}</math>-Hall subgroup and also a 5-complement.		* [[A4 in A5]]: The subgroup has order 12 and index 5 in a group of order 60. It is a <math>\{ 2,3 \}</math>-Hall subgroup and also a 5-complement.
	* [[S4 in S5]]: The subgroup has order 24 and index 5 in a group of order 120. It is a <math>\{ 2,3 \}</math>-Hall subgroup and also a 5-complement.		* [[S4 in S5]]: The subgroup has order 24 and index 5 in a group of order 120. It is a <math>\{ 2,3 \}</math>-Hall subgroup and also a 5-complement.

			Here is a list of examples:

			{{#ask: [[satisfies property::Hall subgroup]][[Category:Particular subgroups]]\|?group part\|?subgroup part\|?quotient part}}

	{{semibasicdef}}		{{semibasicdef}}

Vipul at 00:43, 23 March 2012

2012-03-23T00:43:47Z

@@ Line 17: / Line 17: @@
 We sometimes use the notation <math>\pi'</math> to refer to the complement of <math>\pi</math> in the set of prime numbers.
-Note a few things regarding this definition:
+===Relation between order and prime set specification===
 * The order of a Hall <math>\pi</math>-subgroup of <math>G</math> depends only on the prime set <math>\pi</math> and on the order of <math>G</math>. In particular, for fixed <matH>\pi</math>, all Hall <math>\pi</math>-subgroups have the same order.
 * As far as the definition of Hall <math>\pi</matH>-subgroup of <math>G</matH> is concerned, we ''only'' care about the intersection of <matH>\pi</math> with the set of prime divisors of the order of <math>G</math>. Adding or removing primes that do not divide the order of <math>G</math> does not affect the notion of Hall <math>\pi</math>-subgroup.
 {{semibasicdef}}
 {{subgroup property}}
 [[importance rank::2| ]]
@@ Line 39: / Line 59: @@
 * [[Sylow's theorem]] states for Sylow subgroups (Hall subgroups corresponding to a single prime), the existence, conjugacy, and domination conditions hold in ''all'' [[finite group]]s, not just in finite solvable groups.
 * [[Nilpotent Hall subgroups of same order are conjugate]]
 ==Relation with other properties==

Vipul at 00:33, 23 March 2012

2012-03-23T00:33:59Z

@@ Line 26: / Line 26: @@
 {{subgroup property (finite groups)}}
 [[importance rank::2| ]]
 ==Relation with other properties==

Vipul at 00:19, 23 March 2012

2012-03-23T00:19:16Z

← Older revision		Revision as of 00:19, 23 March 2012
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	~~{{semibasicdef}}~~		==Definition==
	~~{{subgroup property}}~~
	{{subgroup ~~property (finite groups)}}~~		===Definition without prime set specification===
	[[~~importance rank::2\|~~ ]]
	~~==Origin==~~		A [[subgroup]] <math>H</math> of a [[finite group]] <math>G</math> is termed a '''Hall subgroup''' if it satisfies the following equivalent conditions:

	The ~~notion~~ of ~~Hall subgroup was introduced by~~ [[~~Philip Hall~~]] ~~who studied their properties~~ and ~~proved~~ the ~~theorem that a group is solvable if and only if it has Hall subgroups~~ of ~~all possible orders~~.		* The [[order of a group\|order]] of <math>H</math> is relatively prime to the [[index of a subgroup\|index]] of <matH>H</math> in <matH>G</math>.
			* For any prime number <math>p</math> dividing the order of <math>G</math>, <math>p</math> divides ''exactly one'' of the two numbers: the order of <math>H</math> and the index of <math>H</math> in <math>G</math>.

	==Definition==		===Definition with prime set specification===

	~~===Symbol~~-~~free definition===~~		Suppose <math>\pi</math> is a set of [[prime number]]s and <math>G</math> is a [[finite group]]. A [[subgroup]] <math>H</math> of <math>G</math> is termed a <math>\pi</math>-'''Hall subgroup''' or '''Hall <math>\pi</math>-subgroup''' if it satisfies the following equivalent conditions:

	~~A subgroup of a finite group is termed a '''Hall subgroup''' if its~~ [[order of a group\|order]] and [[index of a subgroup\|index]] are ~~coprime~~.		# All the primes dividing the [[order of a group\|order]] of <math>H</math> are in the prime set <math>\pi</math> and all the primes dividing the [[index of a subgroup\|index]] of <math>H</math> in <math>G</math> are outside the prime set <math>\pi</math>.
			# The [[order of a group\|order]] of <math>G</math> is the unique largest divisor of the order of <math>G</math> that has the property that all its prime divisors are in <math>\pi</math>. In other words, it is the <math>\pi</math>-part of the order of <math>G</math>.

	We ~~also have a notion~~ of ~~[[Hall subgroup (profinite groups)\|Hall subgroup~~ in ~~a profinite group]] which generalizes~~ the ~~above notion~~ of ~~Hall subgroup~~.		We sometimes use the notation <math>\pi'</math> to refer to the complement of <math>\pi</math> in the set of prime numbers.

	~~===Definition with symbols===~~		Note a few things regarding this definition:

	A subgroup <math>H</math> of ~~a finite group~~ <math>G</math> ~~is termed a~~ Hall ~~subgroup if~~ the order ~~of H (viz~~ the ~~cardinality~~ of <math>H</math> ~~as a set)~~ is ~~coprime to~~ the ~~index~~ of <~~math~~>H</math> ~~(viz~~ the ~~number~~ of ~~cosets~~ of <math>H</math> in <math>G</math>).		* The order of a Hall <math>\pi</math>-subgroup of <math>G</math> depends only on the prime set <math>\pi</math> and on the order of <math>G</math>. In particular, for fixed <matH>\pi</math>, all Hall <math>\pi</math>-subgroups have the same order.
			* As far as the definition of Hall <math>\pi</matH>-subgroup of <math>G</matH> is concerned, we ''only'' care about the intersection of <matH>\pi</math> with the set of prime divisors of the order of <math>G</math>. Adding or removing primes that do not divide the order of <math>G</math> does not affect the notion of Hall <math>\pi</math>-subgroup.

	~~Equivalently, <math>H</math> is a Hall~~ subgroup ~~if for any prime dividing the order of <math>G</math>, either the prime is ''fully'' inside the order of <math>H</math> or ''fully'' inside the index of <math>H</math>.~~		{{semibasicdef}}
			{{subgroup property}}
			{{subgroup property (finite groups)}}
			[[importance rank::2\| ]]

	==Relation with other properties==		==Relation with other properties==
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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::Sylow subgroup]]		{\| class="sortable" border="1"
	* [[Weaker than::~~Sylow~~ complement]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Weaker than::~~Normal~~ Hall subgroup]]		\|-
	* [[Weaker than::Normal Sylow subgroup]]		\| [[Weaker than::Sylow subgroup]] \|\| Hall subgroup for a single prime, i.e., [[finite p-group]] whose index is relatively prime to <math>p</math> for some prime <math>p</math> \|\| \|\| \|\| {{intermediate notions short\|Hall subgroup\|Sylow subgroup}}
	* [[Weaker than::Hall retract]]		\|-
	* [[Weaker than::Sylow retract]]		\| [[Weaker than::p-complement]] \|\| Hall <math>p'</math>-subgroup, i.e., Hall subgroup whose index is a prime power \|\| \|\| \|\| {{intermediate notions short\|Hall subgroup\|p-complement}}
	* [[Weaker than::~~Nilpotent~~ Hall subgroup]]		\|-
	* [[Weaker than::~~Order~~-dominating Hall subgroup]]		\| [[Weaker than::normal Hall subgroup]] \|\| Hall subgroup that is also a [[normal subgroup]] \|\| \|\| \|\| {{intermediate notions short\|Hall subgroup\|normal Hall subgroup}}
	* [[Weaker than::~~Order~~-conjugate Hall subgroup]]		\|-
	* [[Weaker than::~~Isomorph~~-conjugate Hall subgroup]]		\| [[Weaker than::Normal Sylow subgroup]] \|\| Sylow subgroup that is also a [[normal subgroup]] \|\| \|\| \|\| {{intermediate notions short\|Hall subgroup\|normal Sylow subgroup}}
	* [[Weaker than::~~Pronormal~~ Hall subgroup]]		\|-
			\| [[Weaker than::Hall retract]] \|\| Hall subgroup that is also a [[retract]], i.e., it has a [[normal complement]]. Note that the normal complement must also be a [[Hall subgroup]] for the complementary set of primes \|\| \|\| \|\| {{intermediate notions short\|Hall subgroup\|Hall retract}}
			\|-
			\| [[Weaker than::Sylow retract]] \|\| Sylow subgroup that is also a retract, i.e., <math>p</math>-Sylow subgroup in a group that has a [[normal p-complement]] \|\| \|\| \|\| {{intermediate notions short\|Hall subgroup\|Sylow retract}}
			\|-
			\| [[Weaker than::nilpotent Hall subgroup]] \|\| Hall subgroup that is also a [[nilpotent group]] \|\| \|\| \|\| {{intermediate notions short\|Hall subgroup\|nilpotent Hall subgroup}}
			\|-
			\| [[Weaker than::order-dominating Hall subgroup]] \|\| Hall subgroup that is also an [[order-dominating subgroup]], i.e., any subgroup of the whole group whose order divides it is conjugate to a subgroup of it \|\| \|\| [[Hall not implies order-dominating]] \|\| {{intermediate notions short\|Hall subgroup\|order-dominating Hall subgroup}}
			\|-
			\| [[Weaker than::order-conjugate Hall subgroup]] \|\| Hall subgroup that is also an [[order-conjugate subgroup]], i.e., all Hall subgroups of that order are [[conjugate subgroups]] \|\| \|\| [[Hall not implies order-conjugate]] \|\| {{intermediate notions short\|Hall subgroup\|order-conjugate Hall subgroup}}
			\|-
			\| [[Weaker than::isomorph-conjugate Hall subgroup]] \|\| Hall subgroup that is also an [[isomorph-conjugate subgroup]], i.e., it is [[conjugate subgroups\|conjugate]] to all isomorphic subgroups \|\| \|\| [[Hall not implies isomorph-conjugate]] \|\| {{intermediate notions short\|Hall subgroup\|isomorph-conjugate Hall subgroup}}
			\|-
			\| [[Weaker than::pronormal Hall subgroup]] \|\| Hall subgroup that is also a [[pronormal subgroup]] \|\| \|\| \|\| {{intermediate notions short\|Hall subgroup\|pronormal Hall subgroup}}
			\|}

	===Weaker properties===		===Weaker properties===
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	{{proofat\|[[Hall does not satisfy transfer condition]]}}		{{proofat\|[[Hall does not satisfy transfer condition]]}}

			==History==

			The notion of Hall subgroup was introduced by [[Philip Hall]] who studied their properties and proved the theorem that a group is solvable if and only if it has Hall subgroups of all possible orders. (see [[ECD condition for pi-subgroups in finite solvable groups]] and [[Hall's theorem]]).

Vipul at 02:59, 17 December 2011

2011-12-17T02:59:37Z

← Older revision		Revision as of 02:59, 17 December 2011
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	{{subgroup property}}		{{subgroup property}}
	{{subgroup property (finite groups)}}		{{subgroup property (finite groups)}}
			[[importance rank::2\| ]]
	==Origin==		==Origin==