# Fully invariant not implies normal in loops

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subloop properties. That is, it states that every subloop satisfying the first subloop property (i.e., fully invariant subloop) neednotsatisfy the second subloop property (i.e., normal subloop)

View a complete list of subloop property non-implications | View a complete list of subloop property implications

Get more facts about fully invariant subloop|Get more facts about normal subloop

## Statement

It is possible to have a fully invariant subloop of a loop that is *not* a normal subloop.

## Related facts

- Fully invariant not implies Lagrange-like
- Characteristic not implies normal in loops
- Normal implies Lagrange-like

## Proof

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

Let be a loop with elements and operation as follows:

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:

Suppose is the subloop of . Then:

- is a fully invariant subloop of : In fact, it is the only proper nontrivial subloop, and also the only subloop of order two, so its image under any endomorphism must be either itself or the trivial subloop.
- is not a normal subloop of : For instance, .