Group implies quasigroup

Statement

Verbal statement

Any group is a quasigroup.

Statement with symbols

Let $G$ be a group, and $a,b\in G$ be (not necessarily distinct) elements. Then, there exist unique $x,y\in G$ satisfying $ax=b$ and $ya=b$ respectively.

Definitions used

Quasigroup

Further information: Quasigroup

A magma $(S,*)$ (a set $S$ with binary operation $*$) is termed a quasigroup if for any $a,b\in S$, there exist unique $x,y\in S$ such that $a*x=y*a=b$.

Proof

Given: A group $G$, elements $a,b\in G$

To prove: There exist unique solutions to $ax=b$ and $ya=b$

Proof: We have:

$ax=b\implies a^{-1}(ax)=a^{-1}b\implies x=a^{-1}b$

Conversely:

$x=a^{-1}b\implies ax=a(a^{-1}b)\implies ax=b$

Thus:

$ax=b\iff x=a^{-1}b$

So, $ax=b$ has a unique solution.

Similarly:

$ya=b\implies (ya)a^{-1}=ba^{-1}\implies y=ba^{-1}$

Conversely:

$y=ba^{-1}\implies ya=(ba^{-1})a\implies ya=b$

Thus:

$ya=b\iff y=ba^{-1}$

So, $ya=b$ as a unique solution.