Complemented central factor: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Weaker than::direct factor]] || factor in an [[internal direct product]] || [[direct factor implies complemented central factor]]|[[complemented central factor not implies direct factor]] || {{intermediate notions short|complemented central factor|direct factor}} | |||
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===Weaker properties=== | ===Weaker properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Stronger than::complemented normal subgroup]] || normal subgroup with a permutable complement || || || {{intermediate notions short|complemented normal subgroup|complemented central factor}} | |||
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| [[Stronger than::complemented transitively normal subgroup]] || [[transitively normal subgroup]] with a permutable complement || || || {{intermediate notions short|complemented transitively normal subgroup|complemented central factor}} | |||
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| [[Stronger than::permutably complemented subgroup]] || has a [[permutable complements|permutable complement]] || (via complemented normal) || (via complemented normal) || {{intermediate notions short|permutably complemented subgroup|complemented central factor}} | |||
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| [[Stronger than::lattice-complemented subgroup]] || has a lattice complement || (via permutably complemented) || (via lattice-complemented) || {{intermediate notions short|lattice-complemented subgroup|complemented central factor}} | |||
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| [[Stronger than::central factor]] || product with centralizer is whole group || (by definition) || [[central factor not implies complemented]] || {{intermediate notions short|central factor|complemented central factor}} | |||
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| [[Stronger than::normal subgroup]] || invariant under [[inner automorphism]]s || (via central factor, also via complemented normal) || (via central factor, also via complemented normal) || {{intermediate notions short|normal subgroup|complemented central factor}} | |||
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==Metaproperties== | ==Metaproperties== | ||
Revision as of 04:50, 20 February 2013
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: permutably complemented subgroup and central factor
View other subgroup property conjunctions | view all subgroup properties
Definition
A subgroup of a group is termed a complemented central factor or split central factor if it satisfies the following equivalent conditions:
- It is a permutably complemented subgroup as well as a central factor of the whole group.
- It is a complemented normal subgroup as well as a central factor of the whole group.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| direct factor | factor in an internal direct product | direct factor implies complemented central factor|complemented central factor not implies direct factor | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| complemented normal subgroup | normal subgroup with a permutable complement | |FULL LIST, MORE INFO | ||
| complemented transitively normal subgroup | transitively normal subgroup with a permutable complement | |FULL LIST, MORE INFO | ||
| permutably complemented subgroup | has a permutable complement | (via complemented normal) | (via complemented normal) | |FULL LIST, MORE INFO |
| lattice-complemented subgroup | has a lattice complement | (via permutably complemented) | (via lattice-complemented) | |FULL LIST, MORE INFO |
| central factor | product with centralizer is whole group | (by definition) | central factor not implies complemented | |FULL LIST, MORE INFO |
| normal subgroup | invariant under inner automorphisms | (via central factor, also via complemented normal) | (via central factor, also via complemented normal) | |FULL LIST, MORE INFO |
Metaproperties
Trimness
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties
Both the whole group and the trivial subgroup are complemented central factors.
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition
If is a complemented central factor of a group , then is also a complemented central factor in any intermediate subgroup . This follows because both the property of being a permutably complemented subgroup and the property of being a central factor satisfy the intermediate subgroup condition. For full proof, refer: Permutably complemented satisfies intermediate subgroup condition, Central factor satisfies intermediate subgroup condition