Lagrange-like subloop
This article defines a property that can be evaluated for a subloop of a loop| View other such properties
Definition
A subloop of a finite loop is termed a Lagrange-like subloop if the order (i.e., the number of elements) of the subloop divides the order of the loop.
Relation with other properties
- Algebra loop satisfying the weak Lagrange property is a finite algebra loop in which every subloop is Lagrange-like.
- Algebra loop satisfying the strong Lagrange property is a finite algebra loop in which every subloop satisfies the weak Lagrange property.
Stronger properties
All the properties below are stronger conditional to the ambient loop being finite.
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| direct factor of a loop | |FULL LIST, MORE INFO | |||
| central factor of a loop | |FULL LIST, MORE INFO | |||
| central subloop | |FULL LIST, MORE INFO | |||
| cocentral subloop | |FULL LIST, MORE INFO | |||
| normal subloop | normal implies Lagrange-like | |FULL LIST, MORE INFO | ||
| subnormal subloop | |FULL LIST, MORE INFO |