# Characteristic not implies normal in loops

From Groupprops

ANALOGY BREAKDOWN: This is the breakdown of the analogue in algebra loops of a fact encountered in group. The old fact is: characteristic implies normal.

View other analogue breakdowns of characteristic implies normal|View other analogue breakdowns from group to algebra loop

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., characteristic 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 characteristic subloop|Get more facts about normal subloop

## Statement

### Verbal statement

A characteristic subloop of a loop need not be a normal subloop.

## Related facts

## 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 characteristic subloop of : In fact, it is the only proper nontrivial subloop, and also the only subloop of order two.
- is not a normal subloop of : For instance, .