Loop satisfying the weak Lagrange property

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This article defines a property that can be evaluated for a loop.
View other properties of loops

Definition

An algebra loop is said to satisfy Lagrange's property if it is finite (i.e., its underlying set is finite) and the order (i.e., size) of any subloop of the algebra loop divides the order (i.e., size) of the whole algebra 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 Finite Moufang loop, Loop satisfying the strong Lagrange property|FULL LIST, MORE INFO
Finite Moufang loop every finite Moufang loop satisfies Lagrange's property Loop satisfying the strong Lagrange property|FULL LIST, MORE INFO