Non-power-associative loop of order five

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This article is about a particular loop, viz a loop unique upto isomorphism
View a complete list of particular loops

Definition

This is a loop with five elements. Up to isomorphism, it is the only loop of five elements that is not a power-associative loop.

* 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}

Algebra loop properties

Property Satisfied? Explanation
power-associative loop No \! 3 * (3 * 3) \ne (3 * 3) * 3
flexible loop No \! 3 * (3 * 3) \ne (3 * 3) * 3
alternative loop No \! 3 * (3 * 3) \ne (3 * 3) * 3
Moufang loop No (since it's not alternative)
left Bol loop No (since it's not flexible)
monogenic loop Yes generated by one element

Arithmetic functions

Function Value Explanation
order 5