# Normal subloop

From Groupprops

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)

## Contents

## Definition

### Definition with symbols

A subloop of an algebra loop is said to be **normal** if, for any , the following holds:

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: .
- 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 | |||

central factor of a loop | |FULL LIST, MORE INFO | |||

central subloop | Central factor of a loop|FULL LIST, MORE INFO | |||

cocentral subloop | Central factor of a loop|FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

subnormal subloop | ||||

2-subnormal subloop | ||||

Lagrange-like subloop (in a finite loop) |