Normal subloop

Definition with symbols
A subloop $$N$$ of an algebra loop $$L$$ is said to be normal if, for any $$a,b \in L$$, the following holds:

$$(a * b) * N = a * (b * N) = a * (N * b)$$

Note that the equality of the firsts two is not guaranteed because we do not assume the algebra loop to be associative.

Quotient by a normal subloop
Given a loop, and a normal subloop, we can define a corresponding quotient loop.

Left multiplication group corresponding to a subloop
The following are true:


 * Given a normal subloop, the left multiplication group corresponding to that subloop, is a normal subgroup of the left multiplication group corresponding to the whole algebra loop. Notice that for this, we crucially need the equality of all three parts: $$a * (b * N) = (a * b) * N = a * (N * b)$$.
 * Further, the left multiplication group of the quotient loop equals the quotient of the left multiplication group of the whole loop, by that of the subgroup.