Loop of exponent two
This article defines a property that can be evaluated for a loop.
View other properties of loops
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 |