Normal subloop

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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
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)