Equivalence of definitions of inverse property loop
From Groupprops
This article gives a proof/explanation of the equivalence of multiple definitions for the term inverse property loop
View a complete list of pages giving proofs of equivalence of definitions
Statement
The following are equivalent for a loop .
- Existence of left and right inverses: There exist bijective maps such that .
- Existence of two-sided inverses: There exists a bijective map such that for all .
Related facts
Proof
(1) implies (2)
It suffices to show that ; then we can take the inverse map to equal that.
Given: A loop with a map bijective maps such that .
To prove: For any , .
Proof: Consider the product .
Putting and using the property of , this simplifies to . On the other hand, we know that where is the identity element, so the expression simplifies to . Thus, .
(2) implies (1)
We can take both and as the inverse map.