Central factor of a loop

Definition
A subloop $$S$$ of a loop $$L$$ is termed a central factor if we can write:

$$L = SC_L(S)$$

where $$C_L(S)$$ is the centralizer of $$S$$ in $$L$$, i.e., the elements that both commute and associate with $$S$$.

Incomparable properties

 * Lagrange-like subloop: See central factor not implies Lagrange-like