an associative quasigroup is a group
Proposition 1.
Let $G$ be a set and $\mathrm{\cdot}$ a binary operation^{} on $G$. Write $a\mathit{}b$ for $a\mathrm{\cdot}b$. The following are equivalent^{}:

1.
$(G,\cdot )$ is an associative quasigroup^{}.

2.
$(G,\cdot )$ is an associative loop.

3.
$(G,\cdot )$ is a group.
Proof.
We will prove this in the following direction $(1)\Rightarrow (2)\Rightarrow (3)\Rightarrow (1)$.
 $(1)\Rightarrow (2)$.

Let $x\in G$, and ${e}_{1},{e}_{2}\in G$ such that $x{e}_{1}=x={e}_{2}x$. So $x{e}_{1}^{2}=x{e}_{1}=x$, which shows that ${e}_{1}^{2}={e}_{1}$. Let $a\in G$ be such that ${e}_{1}a=x$. Then ${e}_{2}{e}_{1}a={e}_{2}x=x={e}_{1}a$, so that ${e}_{2}{e}_{1}={e}_{1}={e}_{1}^{2}$, or ${e}_{2}={e}_{1}$. Set $e={e}_{1}$. For any $y\in G$, we have $ey={e}^{2}y$, so $y=ey$. Similarly, $ye=y{e}^{2}$ implies $y=ye$. This shows that $e$ is an identity^{} of $G$.
 $(2)\Rightarrow (3)$.

First note that all of the group axioms are automatically satisfied in $G$ under $\cdot $, except the existence of an (twosided) inverse element, which we are going to verify presently. For every $x\in G$, there are unique elements $y$ and $z$ such that $xy=zx=e$. Then $y=ey=(zx)y=z(xy)=ze=z$. This shows that $x$ has a unique twosided inverse^{} ${x}^{1}:=y=z$. Therefore, $G$ is a group under $\cdot $.
 $(3)\Rightarrow (1)$.

Every group is clearly a quasigroup, and the binary operation is associative.
This completes^{} the proof. ∎
Remark. In fact, if $\cdot $ on $G$ is flexible, then every element in $G$ has a unique inverse: for $z(xz)=(zx)z=ez=z=ze$, so by left division (by $z$), we get $xz=e=xy$, and therefore $z=y$, again by left division (by $x$). However, $G$ may no longer be a group, because associativity may longer hold.
Title  an associative quasigroup is a group 

Canonical name  AnAssociativeQuasigroupIsAGroup 
Date of creation  20130322 18:28:50 
Last modified on  20130322 18:28:50 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Derivation 
Classification  msc 20N05 
Related topic  Group 