# Lagrange-like subloop

From Groupprops

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 | Central factor of a loop|FULL LIST, MORE INFO | |||

central factor of a loop | |FULL LIST, MORE INFO | |||

central subloop | Central factor of a loop|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 |