# Non-power-associative loop of order five

From Groupprops

This article is about a particular loop, viz a loop unique upto isomorphism

View a complete list of particular loops

## Definition

This is a loop with five elements. Up to isomorphism, it is the only loop of five elements that is not a power-associative loop.

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

1 | 1 | 2 | 3 | 4 | 5 |

2 | 2 | 1 | 5 | 3 | 4 |

3 | 3 | 5 | 4 | 2 | 1 |

4 | 4 | 3 | 1 | 5 | 2 |

5 | 5 | 4 | 2 | 1 | 3 |

In other words, this is the algebra loop corresponding to the Latin square:

## Algebra loop properties

Property | Satisfied? | Explanation |
---|---|---|

power-associative loop | No | |

flexible loop | No | |

alternative loop | No | |

Moufang loop | No | (since it's not alternative) |

left Bol loop | No | (since it's not flexible) |

monogenic loop | Yes | generated by one element |

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order | 5 |