Finite elliptic 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:: | | [[Weaker than::finite characteristic subgroup]] || finite and a [[characteristic subgroup]] || || || | ||
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| [[Weaker than:: | | [[Weaker than::finite normal subgroup]] || finite and a [[normal subgroup]] || || || | ||
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| [[Weaker than:: | | [[Weaker than::finite permutable subgroup]] || finite and a [[permutable subgroup]] || || || | ||
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| [[Weaker than::subgroup of finite group]] || the whole group is finite || || || | |||
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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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| [[Stronger than:: | | [[Stronger than::finite subgroup]] || || || || | ||
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| [[Stronger than:: | | [[Stronger than::elliptic subgroup]] || || || || | ||
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| [[Stronger than:: | | [[Stronger than::join-transitively finite subgroup]] || || || || | ||
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Latest revision as of 04:07, 9 March 2020
This article describes a property that arises as the conjunction of a subgroup property: elliptic subgroup with a group property (itself viewed as a subgroup property): finite group
View a complete list of such conjunctions
Definition
A subgroup of a group is termed a finite elliptic subgroup if it is a finite subgroup (i.e., is a finite group in its own right) and is an elliptic subgroup of the whole group.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finite characteristic subgroup | finite and a characteristic subgroup | |||
| finite normal subgroup | finite and a normal subgroup | |||
| finite permutable subgroup | finite and a permutable subgroup | |||
| subgroup of finite group | the whole group is finite |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finite subgroup | ||||
| elliptic subgroup | ||||
| join-transitively finite subgroup |