# Nonempty associative quasigroup equals group

## Contents

## History

The fact that a (nonempty) associative quasigroup is the same thing as a group is more than a mere curiosity. Many of the initial definitions of group, including Cayley's first attempted definition in 1854 and Weber's definition in 1882, defined a group as an associative quasigroup, without explicitly mentioning identity elements and inverses.

`Further information: History of groups`

## Statement

Let be a nonempty magma (set with binary operation) satisfying the following two conditions:

- is a quasigroup under , i.e., for any , there exist unique such that
- is associative, or is a semigroup under , i.e., for any , we have

Then is a group under .

Conversely, any group is an associative quasigroup. Associativity is part of the definition, and the quasigroup part is also direct. `For full proof, refer: Group implies quasigroup`

## Related facts

- Nonempty alternative quasigroup equals alternative loop: Here, alternativity, which is a weaker condition than associativity, also forces a nonempty quasigroup to have a neutral element.

## Facts used

## Proof

Associativity is given to us, so we need to find an identity element (neutral element for the multiplication) and inverses.

### Finding the identity element (or neutral element)

Since is nonempty, we can pick . By the quasigroup condition, find such that .

Now, pick any . By the quasigroup condition again, there exists a unique element such that . Plugging in, we get:

Thus, is a right neutral element (or right identity) for the multiplication .

By a similar procedure, we can find a left neutral element. Further, since any left and right neutral element are equal, we see that must be a two-sided identity element for .

### Finding inverses

If is the identity element, then for any , we can find such that . Thus, has both a left inverse and a right inverse. By associativity, any left and right inverse must be equal, so has a two-sided inverse. Thus, every element has a two-sided inverse, so every element is invertible, and is thus a group.