Associative quasigroups: They are groups

Explain associative quasigroups

Sure. Here’s the picture in full:

1. Definitions

• A quasigroup $(Q,*)$ is a set with a binary operation $*$ such that for every $a,b\in Q$ the equations $a*x=b$ and $y*a=b$ have unique solutions $x,y\in Q$. Equivalently, left and right multiplication by any fixed element are bijections. (Wikipedia)

• A loop is a quasigroup that has an identity element $e$ with $e*a = a*e = a$ for all $a$. (Wikipedia)

• An associative quasigroup is a quasigroup whose operation is associative: $(a*b)*c = a*(b*c)$ for all $a,b,c\in Q$. (groupprops.subwiki.org)

2. Main fact

Any nonempty associative quasigroup is a group. In other words, associativity together with the quasigroup division property forces the existence of a two-sided identity and two-sided inverses, so $(Q,*)$ satisfies all the group axioms. Conversely, every group is trivially an associative quasigroup. (groupprops.subwiki.org, Wikipedia)

(Minor technical note: the empty set can be vacuously an associative quasigroup under some conventions, but it lacks an identity element and so is not a group in the usual sense. The standard theorem refers to nonempty associative quasigroups. ) (Mathematics Stack Exchange)

3. Sketch of the proof (from the standard argument): (groupprops.subwiki.org)

Let $(Q,*)$ be an associative quasigroup and pick some $a\in Q$ (so $Q$ is assumed nonempty).

• By the quasigroup property, there is a unique $e\in Q$ with $a*e = a$. For an arbitrary $b\in Q$, pick $y$ with $y*a = b$; then

  $$ b*e = (y*a)*e = y*(a*e) = y*a = b $$

  using associativity. Thus $e$ is a right identity for all elements. Likewise one constructs a left identity, and uniqueness of left/right neutral elements forces them to coincide, giving a two-sided identity $e$.

• Now with identity $e$ in hand, for any $a\in Q$ the quasigroup property gives $x,y$ such that $a*x = e$ and $y*a = e$. Associativity implies any left and right inverse coincide, so each element has a two-sided inverse.

Hence $Q$ has an identity and every element is invertible, and the operation is associative: it is a group. (groupprops.subwiki.org)

4. Consequences / interpretation

So “associative quasigroup” (with nonempty underlying set) is just another way to axiomatize a group: one can define a group as an associative quasigroup, without separately postulating an identity and inverses. Historically some early definitions of groups were phrased that way. (groupprops.subwiki.org)

5. Examples

• Any group $(G,\cdot)$ is an associative quasigroup because it has associativity and the division property (invertibility gives unique solutions to $a*x=b$ and $y*a=b$). (Wikipedia)
• There are no new “non-group” examples: if you assume associativity in a quasigroup, you’re forced into a group structure. (groupprops.subwiki.org)

If you’d like, I can also give variants (e.g., what weaker associativity-like identities still force a quasigroup to be a loop/group) or examples of nonassociative quasigroups to contrast. Would you like that?