Vipul at 13:16, 26 September 2008

2008-09-26T13:16:42Z

← Older revision		Revision as of 13:16, 26 September 2008
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	{{subgroup property conjunction\|Hall subgroup\|direct factor}}		{{subgroup property conjunction\|Hall subgroup\|direct factor}}
	{{subgroup property conjunction\|Hall subgroup\|central factor}}		{{subgroup property conjunction\|Hall subgroup\|central factor}}
	~~{{subgroup property conjunction\|Hall subgroup\|conjugacy-closed normal subgroup}}~~

	==Definition==		==Definition==
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	# It is a [[Hall subgroup]] and is also a [[direct factor]].		# It is a [[Hall subgroup]] and is also a [[direct factor]].
	# It is a [[normal Hall subgroup]] and possesses a normal complement, i.e., is a [[retract]].		# It is a [[normal Hall subgroup]] and possesses a normal complement, i.e., is a [[retract]].
	~~# It is a [[Hall subgroup]] and is also a [[conjugacy-closed normal subgroup]].~~
	# It is a [[Hall subgroup]] and is also a [[central factor]] of the whole group.		# It is a [[Hall subgroup]] and is also a [[central factor]] of the whole group.

	===Equivalence of definitions===		===Equivalence of definitions===

	{{proofat\|~~[[Conjugacy-closed normal Hall implies central factor]],~~ [[Hall and central factor implies direct factor]]}}		{{proofat\|[[Hall and central factor implies direct factor]]}}

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

Vipul: New page: {{subgroup property conjunction|Hall subgroup|direct factor}} {{subgroup property conjunction|Hall subgroup|central factor}} {{subgroup property conjunction|Hall subgroup|conjugacy-closed ...

2008-09-26T12:57:54Z

New page: {{subgroup property conjunction|Hall subgroup|direct factor}} {{subgroup property conjunction|Hall subgroup|central factor}} {{subgroup property conjunction|Hall subgroup|conjugacy-closed ...

New page

{{subgroup property conjunction|Hall subgroup|direct factor}}
{{subgroup property conjunction|Hall subgroup|central factor}}
{{subgroup property conjunction|Hall subgroup|conjugacy-closed normal subgroup}}

==Definition==

A [[subgroup]] of a [[finite group]] is termed a '''Hall direct factor''' if it satisfies the following equivalent conditions:

# It is a [[Hall subgroup]] and is also a [[direct factor]].
# It is a [[normal Hall subgroup]] and possesses a normal complement, i.e., is a [[retract]].
# It is a [[Hall subgroup]] and is also a [[conjugacy-closed normal subgroup]].
# It is a [[Hall subgroup]] and is also a [[central factor]] of the whole group.

===Equivalence of definitions===

{{proofat|[[Conjugacy-closed normal Hall implies central factor]], [[Hall and central factor implies direct factor]]}}

==Relation with other properties==

===Stronger properties===

* [[Weaker than::Sylow direct factor]]

===Weaker properties===

* [[Stronger than::Normal Hall subgroup]]
* [[Stronger than::Fully characteristic subgroup]]
* [[Stronger than::Fully characteristic direct factor]]
* [[Stronger than::Direct factor]]
* [[Stronger than::Central factor]]
* [[Stronger than::Conjugacy-closed normal subgroup]]
* [[Stronger than::Conjugacy-closed Hall subgroup]]

==Metaproperties==

{{transitive}}

{{trim}}

{{intsubcondn}}

Hall direct factor - Revision history

Vipul at 13:16, 26 September 2008

Vipul: New page: {{subgroup property conjunction|Hall subgroup|direct factor}} {{subgroup property conjunction|Hall subgroup|central factor}} {{subgroup property conjunction|Hall subgroup|conjugacy-closed ...