Normal subloop

From Groupprops

This article defines a property that can be evaluated for a subloop of an algebra loop
View other such properties

ANALOGY: This is an analogue in algebra loop of a property encountered in group. Specifically, it is a subloop property analogous to the subgroup property: normal subgroup
View other analogues of normal subgroup | View other analogues in algebra loops of subgroup properties (OR, View as a tabulated list)

Definition

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.

Facts

Quotient by a normal subloop

Given a loop, and a normal subloop, we can define a corresponding quotient loop. Fill this in later

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.