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:

$ax=b⟹a^{-1}(ax)=a^{-1}b⟹x=a^{-1}b$

Conversely:

$x=a^{-1}b⟹ax=a(a^{-1}b)⟹ax=b$

Thus:

$ax=b⟺x=a^{-1}b$

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

Similarly:

$ya=b⟹(ya)a^{-1}=b{a}^{-1}⟹y=b{a}^{-1}$

Conversely:

$y=b{a}^{-1}⟹ya=(b{a}^{-1})a⟹ya=b$

Thus:

$ya=b⟺y=b{a}^{-1}$

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