Powering-invariant characteristic subgroup: Difference between revisions
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| [[Weaker than::characteristic subgroup of abelian group]] || || [[characteristic subgroup of abelian group is powering-invariant]] || || {{intermediate notions short|powering-invariant characteristic subgroup|characteristic subgroup of abelian group}} | | [[Weaker than::characteristic subgroup of abelian group]] || || [[characteristic subgroup of abelian group is powering-invariant]] || || {{intermediate notions short|powering-invariant characteristic subgroup|characteristic subgroup of abelian group}} | ||
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| [[Weaker than::complemented characteristic subgroup]] || || (via quotient-powering-invariant characteristic) || || | | [[Weaker than::complemented characteristic subgroup]] || || (via quotient-powering-invariant characteristic) || || {{intermediate notions short|powering-invariant characteristic subgroup|complemented characteristic subgroup}} | ||
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| [[Weaker than::finite characteristic subgroup]] || || (via quotient-powering-invariant characteristic)|| || | | [[Weaker than::finite characteristic subgroup]] || || (via quotient-powering-invariant characteristic)|| || {{intermediate notions short|powering-invariant characteristic subgroup|finite characteristic subgroup}} | ||
|- | |- | ||
| [[Weaker than::characteristic subgroup of finite index]] || || (via quotient-powering-invariant characteristic)|| || | | [[Weaker than::characteristic subgroup of finite index]] || || (via quotient-powering-invariant characteristic)|| || {{intermediate notions short|powering-invariant characteristic subgroup|characteristic subgroup of finite index}} | ||
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===Weaker properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::powering-invariant normal subgroup]] || [powering-invariant subgroup]] as well as a [[normal subgroup]] || follows from [[characteristic implies normal]] || follows from finite examples for [[normal not implies characteristic]] || {{intermediate notions short|powering-invariant normal subgroup|powering-invariant characteristic subgroup}} | |||
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| [[Stronger than::powering-invariant subgroup]] || || || || {{intermediate notions short|powering-invariant subgroup|powering-invariant characteristic subgroup}} | |||
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| [[Stronger than::characteristic subgroup]] || || || || {{intermediate notions short|characteristic subgroup|powering-invariant characteristic subgroup}} | |||
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| [[Stronger than::normal subgroup]] || || || || {{intermediate notions short|normal subgroup|powering-invariant characteristic subgroup}} | |||
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Latest revision as of 20:45, 16 February 2013
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: powering-invariant subgroup and characteristic subgroup
View other subgroup property conjunctions | view all subgroup properties
Definition
A subgroup of a group is termed a powering-invariant characteristic subgroup if it is both a powering-invariant subgroup and a characteristic subgroup.
Note that characteristic not implies powering-invariant. Also, a non-characteristic subgroup of a finite group is powering-invariant and not characteristic. Hence, neither property implies the other, so the conjunction is strictly stronger than both.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| quotient-powering-invariant characteristic subgroup | ||||
| characteristic subgroup of abelian group | characteristic subgroup of abelian group is powering-invariant | |FULL LIST, MORE INFO | ||
| complemented characteristic subgroup | (via quotient-powering-invariant characteristic) | |FULL LIST, MORE INFO | ||
| finite characteristic subgroup | (via quotient-powering-invariant characteristic) | |FULL LIST, MORE INFO | ||
| characteristic subgroup of finite index | (via quotient-powering-invariant characteristic) | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| powering-invariant normal subgroup | [powering-invariant subgroup]] as well as a normal subgroup | follows from characteristic implies normal | follows from finite examples for normal not implies characteristic | |FULL LIST, MORE INFO |
| powering-invariant subgroup | |FULL LIST, MORE INFO | |||
| characteristic subgroup | |FULL LIST, MORE INFO | |||
| normal subgroup | |FULL LIST, MORE INFO |