Conjugate-comparable subgroup: 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 | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
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| [[Weaker than::Normal subgroup]] || || | | [[Weaker than::Automorph-comparable subgroup]] || comparable to all its [[automorphic subgroups]] || || || {{intermediate notions short|conjugate-comparable subgroup|automorph-comparable subgroup}} | ||
| [[Weaker than::Normal subgroup]] || equal to each conjuate subgroup || equal things can be compared || [[conjugate-comparable not implies normal]] || {{intermediate notions short|conjugate-comparable subgroup|normal subgroup}} | |||
|} | |} | ||
==Facts== | |||
* [[Automorph-comparable of normal implies conjugate-comparable]] | |||
* [[Left residual of conjugate-comparable by normal is automorph-comparable]] | |||
Revision as of 22:10, 4 May 2010
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 a conjugate-comparable subgroup if it is comparable with each of its conjugate subgroups, in other words, every conjugate subgroup to it either contains it or is contained in it.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions | |||||
|---|---|---|---|---|---|---|---|---|---|
| Automorph-comparable subgroup | comparable to all its automorphic subgroups | |FULL LIST, MORE INFO | Normal subgroup | equal to each conjuate subgroup | equal things can be compared | conjugate-comparable not implies normal | |FULL LIST, MORE INFO |