# Normal subloop

This article defines a property that can be evaluated for a subloop of a loop| View other such properties
ANALOGY: This is an analogue in 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 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. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

### 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.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
direct factor of a loop Central factor of a loop, Right-transitively normal subloop|FULL LIST, MORE INFO