# Central 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 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 of an algebra loop if, for all and , we have:

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 |
---|---|---|---|---|

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

Nuclear subloop | contained in the nucleus | |FULL LIST, MORE INFO |