Group implies quasigroup
Statement with symbols
Let be a group, and be (not necessarily distinct) elements. Then, there exist unique satisfying and respectively.
Further information: Quasigroup
A magma (a set with binary operation ) is termed a quasigroup if for any , there exist unique such that .
Given: A group , elements
To prove: There exist unique solutions to and
Proof: We have:
So, has a unique solution.
So, as a unique solution.