Elliptic subgroup: Difference between revisions
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{{subgroup property}} | {{subgroup property}} | ||
{{finitarily tautological subgroup property}} | |||
==Definition== | ==Definition== | ||
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A subgroup <math>H</math> of a group <math>G</math> is termed '''elliptic''' if for any subgroup <math>K</math> of <math>G</math>, <math>(H,K)</math> form an [[elliptic pair of subgroups]]. In other words, there exists an <math>n</math> such that: | A subgroup <math>H</math> of a group <math>G</math> is termed '''elliptic''' if for any subgroup <math>K</math> of <math>G</math>, <math>(H,K)</math> form an [[elliptic pair of subgroups]]. In other words, there exists an <math>n</math> such that: | ||
<math> | <math>\langle H,K \rangle = (HK)^n := HKHKHK \ldots HK</math> | ||
where each is written <math>n</math> times. | where each is written <math>n</math> times. | ||
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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::normal subgroup]] || || || || {{intermediate notions short|elliptic subgroup|normal subgroup}} | |||
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| [[Weaker than::permutable subgroup]] || permutes with every subgroup || || || {{intermediate notions short|elliptic subgroup|permutable subgroup}} | |||
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| [[Weaker than::subgroup of finite index]] || || || || {{intermediate notions short|elliptic subgroup|subgroup of finite index}} | |||
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Latest revision as of 04:18, 9 March 2020
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]
This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
View other such subgroup properties
Definition
Symbol-free definition
A subgroup of a group is termed elliptic if it forms an elliptic pair of subgroups with every subgroup of the group.
Definition with symbols
A subgroup of a group is termed elliptic if for any subgroup of , form an elliptic pair of subgroups. In other words, there exists an such that:
where each is written times.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| normal subgroup | |FULL LIST, MORE INFO | |||
| permutable subgroup | permutes with every subgroup | |FULL LIST, MORE INFO | ||
| subgroup of finite index | |FULL LIST, MORE INFO |