Loop of order five and exponent two

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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 loop is defined by the multiplication table:

* 1 2 3 4 5
1 1 2 3 4 5
2 2 1 4 5 3
3 3 5 1 2 4
4 4 3 5 1 2
5 5 4 2 3 1


In other words, this is the algebra loop corresponding to the Latin square:

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

Loop properties

Property Satisfied? Explanation
Loop of exponent two Yes Every non-identity element squares to the identity
Power-associative loop Yes Follows since it is a loop of exponent two
Flexible loop Yes In fact, a * (b * a) = (a * b) * a = b for all a,b
Commutative loop No For instance, 2 * 3 \ne 3 * 2
Left alternative loop No For instance, 2 * (2 * 3) \ne (2 * 2) * 3
Right alternative loop No For instance, 2 * (3 * 3) \ne (2 * 3) * 3
Alternative loop No Neither left nor right alternative
Left Bol loop No Not left alternative, so not left Bol
Right Bol loop No Not right alternative, so not right Bol
Moufang loop No Not alternative, so not Moufang
Left nuclear square loop Yes Every square is the identity element, hence in the left nucleus
Middle nuclear square loop Yes Every square is the identity element, hence in the middle nucleus
Right nuclear square loop Yes Every square is the identity element, hence in the right nucleus
LC-loop No Not left alternative, so not LC.
RC-loop No Not right alternative, so not RC.
C-loop No Not alternative, so not C.
Monogenic loop No The subloop generated by every element is a subgroup of order two.