Center of a loop

Definition

Let $L$ be an algebra loop with binary operation $*$. Then, the center of $L$, denoted as $Z(L)$, is defined as the set of all elements $a\in L$ satisfying:

$a*(x*y)=x*(a*y)=(x*a)*y=(x*y)*a\forall x,y\in L$

Facts

Subloop properties satisfied

Property	Meaning	Proof of satisfaction
Normal subloop		center is normal for algebra loops
Characteristic subloop		center is characteristic for algebra loops

Subloop properties not satisfied

Property	Meaning	Proof of satisfaction
Lagrange-like subloop	in a finite loop, means that the order of the subloop divides the order of the loop	center not is Lagrange-like