# Loop of exponent two

This article defines a property that can be evaluated for a loop.

View other properties of loops

## Contents

## Definition

A nontrivial loop is termed a **loop of exponent two** if the square of every element in the loop is the identity element of the loop.

Whether we consider the trivial loop a loop of exponent two is a moot point.

## Examples

An example that is non-abelian (and not a group) is the loop of order five and exponent two.

## Relation with other properties

### Weaker properties

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

Power-associative loop | powers are well-defined, independent of parenthesization | |FULL LIST, MORE INFO | ||

Left nuclear square loop | every square element is in the left nucleus | |FULL LIST, MORE INFO | ||

Middle nuclear square loop | every square element is in the middle nucleus | |FULL LIST, MORE INFO | ||

Right nuclear square loop | every square element is in the right nucleus | |FULL LIST, MORE INFO |