Every finite Moufang loop satisfies the weak Lagrange property

Statement

In terms of the weak Lagrange property

Suppose $(L,*)$ is a finite Moufang loop, i.e., a Moufang loop whose underlying set is finite. Then, $L$ is an algebra loop satisfying the weak Lagrange property, i.e., every subloop of $L$ is a Lagrange-like subloop (?): its order divides the order of the loop.

In terms of the strong Lagrange property

Suppose $(L,*)$ is a finite Moufang loop, i.e., a Moufang loop whose underlying set is finite. Then, $L$ is an algebra loop satisfying the strong Lagrange property, i.e., every subloop of $L$ satisfies the weak Lagrange property.

Facts used

1. Moufang property is subloop-closed

Proof

How the statement for the strong Lagrange property follows from the statement for the weak Lagrange property

If every finite Moufang loop satisfies the weak Lagrange property, then by fact (1), every subloop of a finite Moufang loop also satisfies the weak Lagrange property, and we thus obtain that every finite Moufang loop satisfies the strong Lagrange property.