Central factor of a loop

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: central factor
View other analogues of central factor | View other analogues in loops of subgroup properties (OR, View as a tabulated list)

Definition

A subloop $S$ of a loop $L$ is termed a central factor if we can write:

$L=SC_{L}(S)$

where $C_{L}(S)$ is the centralizer of $S$ in $L$, i.e., the elements that both commute and associate with $S$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
direct factor of a loop				|FULL LIST, MORE INFO
cocentral subloop				|FULL LIST, MORE INFO
central subloop				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Lagrange-like subloop
right-transitively normal subloop
normal subloop				|FULL LIST, MORE INFO

Incomparable properties

• Lagrange-like subloop: See central factor not implies Lagrange-like