# Moufang implies alternative

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two loop properties. That is, it states that every loop satisfying the first loop property (i.e., Moufang loop) must also satisfy the second loop property (i.e., alternative loop)

View all loop property implications | View all loop property non-implications

Get more facts about Moufang loop|Get more facts about alternative loop

## Contents

## Statement

Any Moufang loop is an alternative loop.

## Related facts

### Stronger facts

### Applications

## Definitions used

### Moufang loop

`Further information: Moufang loop`

A Moufang loop is a loop satisfying the following identities for all (where two or more of the could possibly be equal):

### Alternative loop

`Further information: alternative loop`

An alternative loop is a loop satisfying the following two identities for all (where may be equal or distinct):

## Proof

**Given**: A Moufang loop with identity element .

**To prove**: For all (possibly equal), we have (left alternative law) and (right alternative law).

**Proof**: For the left alternative law, set , , and in Moufang's identity (1) given above.

For the right alternative law, set , , and in Moufang's identity (2) given above.