# Central 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 algebra loop of a property encountered in group. Specifically, it is a subloop property analogous to the subgroup property: central subgroup
View other analogues of central subgroup | View other analogues in algebra loops of subgroup properties (OR, View as a tabulated list)

## Definition

A subloop of an algebra loop is termed a central subloop or central subgroup if it is contained in the center of the loop. In other words, a subloop $S$ of an algebra loop $(L,*)$ if, for all $a \in S$ and $x,y \in L$, we have:

$\! x * (y * a) = (x * y) * a = a * (x * y) = (a * x) * y$

Note that any central subloop is an abelian group under the induced multiplication because the multiplication operation on it is associative.

## Relation with other properties

### Weaker properties

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