Characteristic not implies normal in loops

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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) need not satisfy 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 L be a loop with elements 1,2,3,4,5 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:

\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 5 & 3 & 4 \\ 3 & 5 & 4 & 2 & 1 \\ 4 & 3 & 1 & 5 & 2 \\ 5 & 4 & 2 & 1 & 3 \\\end{pmatrix}

Suppose S is the subloop \{ 1,2 \} of L. Then:

  • S is a characteristic subloop of L: In fact, it is the only proper nontrivial subloop, and also the only subloop of order two.
  • S is not a normal subloop of L: For instance, (S * 3) * 3 \ne S * (3 * 3).