# Quasigroup

From Groupprops

This is a variation of group|Find other variations of group | Read a survey article on varying group

QUICK PHRASES: group without identity and associativity, magma with unique left and right quotients

## Contents

## Definition

### Definition with symbols

A **quasigroup** is a magma (set with binary operation) with the following properties:

- For every , there is a unique such that
- For every , there is a unique such that

Note that may be equal or different for a particular choice of and .

Equivalently, a quasigroup is a magma where every element is cancellative and where every element is left-accessible and right-accessible from every other element.

## Relation with other notions

### Stronger notions

- Group:
`For full proof, refer: Group implies quasigroup` - Loop: A loop is a quasigroup with a two-sided multiplicative identity element (or neutral element)

Any nonempty quasigroup that is also a semigroup is a group. `For full proof, refer: Associative quasigroup implies group`

### Weaker notions

A finite magma is a quasigroup if and only if its multiplication table is a Latin square.