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Central factor satisfies intermediate subgroup condition
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about central factor | Get facts that use property satisfaction of central factor | Get facts that use property satisfaction of central factor| Get more facts about intermediate subgroup condition
Contents |
Statement
Verbal statement
A central factor of a group is also a central factor in every intermediate subgroup.
Related facts
Generalizations and other particular cases of the generalizations
- Left-inner implies intermediate subgroup condition
- Left-extensibility-stable implies intermediate subgroup condition
- Normality satisfies intermediate subgroup condition
- Centrality satisfies intermediate subgroup condition
- Cocentrality satisfies intermediate subgroup condition
Related metaproperties for central factor
- Central factor does not satisfy transfer condition
- Central factor satisfies image condition
- Central factor is upper join-closed
Proof
Using function restriction expressions
This subgroup property implication can be proved by using function restriction expressions for the subgroup properties
View other implications proved this way |read a survey article on the topic
This proof method generalizes to the following results: left-inner implies intermediate subgroup condition, left-extensibility-stable implies intermediate subgroup condition
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Proof using product with centralizer definition
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