Loop of exponent two

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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