# Prime order not implies simple for algebra loops

From Groupprops

Template:Algebra loop property non-implication

## Statement

It is possible to have an algebra loop that is a loop of prime order (i.e., its order is a Prime number (?)) but has a proper nontrivial Normal subloop (?), and is hence not a simple loop.

## Proof

`Further information: non-power-associative loop of order five`

Consider the algebra loop of order five with multiplication table:

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:

It is easy to check that the subset is a normal subloop of the loop. In fact, it is precisely the center of the loop, i.e., its elements commute and associate with everything.