Upward-closed characteristic subgroup: Difference between revisions
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
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| [[Weaker than::Cyclic-quotient characteristic subgroup]] || [[characteristic subgroup]], [[quotient group]] is [[cyclic group|cyclic]] || [[cyclic-quotient characteristic implies upward-closed characteristic]] || [[upward-closed characteristic not implies cyclic in finite]] || {{intermediate notions short|upward-closed characteristic subgroup|cyclic-quotient characteristic subgroup}} | | [[Weaker than::Cyclic-quotient characteristic subgroup]] || [[characteristic subgroup]], [[quotient group]] is [[cyclic group|cyclic]] || [[cyclic-quotient characteristic implies upward-closed characteristic]] || [[upward-closed characteristic not implies cyclic-quotient in finite]] || {{intermediate notions short|upward-closed characteristic subgroup|cyclic-quotient characteristic subgroup}} | ||
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| [[Weaker than::Characteristic subgroup of prime index]] || [[characteristic subgroup]] and [[subgroup of prime index]] || || || {{intermediate notions short|upward-closed characteristic subgroup|characteristic subgroup of prime index}} | | [[Weaker than::Characteristic subgroup of prime index]] || [[characteristic subgroup]] and [[subgroup of prime index]] || || || {{intermediate notions short|upward-closed characteristic subgroup|characteristic subgroup of prime index}} | ||
Latest revision as of 01:23, 22 January 2010
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
Definition
A subgroup of a group is termed upward-closed characteristic if, for every subgroup of containing , is also a characteristic subgroup of .
Formalisms
In terms of the upward-closure operator
This property is obtained by applying the upward-closure operator to the property: characteristic subgroup
View other properties obtained by applying the upward-closure operator
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Cyclic-quotient characteristic subgroup | characteristic subgroup, quotient group is cyclic | cyclic-quotient characteristic implies upward-closed characteristic | upward-closed characteristic not implies cyclic-quotient in finite | |FULL LIST, MORE INFO |
| Characteristic subgroup of prime index | characteristic subgroup and subgroup of prime index | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Upward-closed normal subgroup | every subgroup containing it is normal in whole group |