Center of a loop

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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