Prime order not implies simple for algebra loops

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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:

\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}

It is easy to check that the subset \{ 1,2 \} 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.