Loop of order five and exponent two

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, $2*3\neq 3*2$
Left alternative loop	No	For instance, $2*(2*3)\neq (2*2)*3$
Right alternative loop	No	For instance, $2*(3*3)\neq (2*3)*3$
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.