Loop satisfying the weak Lagrange property: Difference between revisions
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{{ | {{loop property}} | ||
==Definition== | ==Definition== | ||
An [[ | An [[finite loop]] is said to satisfy the '''weak property''' if every [[subloop]] is a [[defining ingredient::Lagrange-like subloop]], i.e., the [[defining ingredient::order of a loop|order]] (number of elements) of any subloop divides the order of the loop. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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| [[Weaker than::Finite group]] || || [[Lagrange's theorem]] || || {{intermediate notions short|algebra loop satisfying Lagrange's property|finite group}} | | [[Weaker than::Finite group]] || || [[Lagrange's theorem]] || || {{intermediate notions short|algebra loop satisfying Lagrange's property|finite group}} | ||
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| [[Weaker than::Finite Moufang loop]] || || [[every finite Moufang loop satisfies Lagrange | | [[Weaker than::Finite Moufang loop]] || || [[every finite Moufang loop satisfies the weak Lagrange property]] || || {{intermediate notions short|algebra loop satisfying Lagrange's property|finite Moufang loop}} | ||
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| [[Weaker than::Loop satisfying the strong Lagrange property]] || every subloop satisfies the weak Lagrange property || || || {{intermediate notions short|algebra loop satisfying the weak Lagrange property|algebra loop satisfying the strong Lagrange property}} | |||
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Latest revision as of 16:04, 9 March 2010
This article defines a property that can be evaluated for a loop.
View other properties of loops
Definition
An finite loop is said to satisfy the weak property if every subloop is a Lagrange-like subloop, i.e., the order (number of elements) of any subloop divides the order of the loop.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Finite group | Lagrange's theorem | |FULL LIST, MORE INFO | ||
| Finite Moufang loop | every finite Moufang loop satisfies the weak Lagrange property | |FULL LIST, MORE INFO | ||
| Loop satisfying the strong Lagrange property | every subloop satisfies the weak Lagrange property | |FULL LIST, MORE INFO |