Example answering Questions 5.3 and 5.5 of On core quandles of groups

I describe below an involutive quandle which satisfies the equivalent conditions of Proposition 5.2, equivalently, condition (5.2), of my paper On core quandles of groups, but which cannot be embedded in  Conj(G)  for any group  G.  In view of the second statement of Proposition 6.2 of that paper, such a quandle also cannot be embedded in  Core(H)  for any group  H.  Hence, this example gives negative answers to Questions 5.3 and 5.5 of that paper. 

Let  Q  be an 8-element set with elements named  xiyizij  (i, j ∈ {0, 1}).  Define a binary operation  ◁  on  Q  as follows:  Let each of the subsets  {x0, x1},  {y0, y1},  {z00, z01, z10, z11},  act trivially on itself under  ◁.  Let the action on  {x0, x1}  of every element not in that set interchange  x0  and  x1,  and similarly let the action on  {y0, y1}  of every element not in that set interchange  y0  and  y1.  Finally, let the action of each of the x's on the z's interchange 0 and 1 in the first subscript, leaving the second subscript unchanged, and let the action of each of the y's on the z's interchange 0 and 1 in the second subscript, leaving the first subscript unchanged.  It is straightforward to verify that this is an involutory quandle structure, and satisfies condition (5.2). 

Suppose, now, that this quandle  Q  were a subquandle of  Conj(G)  for some group  G.  We may assume without loss of generality that  G  is generated by the elements of  Q

Note that since  x0  and  x1  have the same action by  ◁  on  Q,  which generates  G  as a group, they must have the same action by conjugation on all elements of  G,  hence they must differ in  G  by a central element  a;  and since the conjugation automorphisms that take  x0  to  x1  also take  x1 = x0 a  to  x0,  we must have  a2 = 1.  Letting  w  denote any of the  yi  or  zij,  the relation  w−1xi w  = xi a  can be rewritten  w = xi−1w  xi a,  whence, using the fact that  a2 = 1,  we get  xi−1w xiwa.  Letting  w = yi  in this relation, we see that  y0  and  y1  differ by a factor of  a.  Similarly, letting  w = zij,  we see that  z0j   and  z1j  differ by a factor of  a

But knowing that  y0  and  y1  differ by a factor of  a,  we can repeat the observations of the above paragraph with  y0  and  y1  in place of  x0  and  x1,  this time concluding (in view of the different action of the  yi  on the  zij )  that  zi0  and  zi1  differ by  a.  Combining these results, we conclude that  z00  and  z11  differ by a factor of  a2 = 1,  i.e., are equal, contradicting our original description of  Q,  and thus showing that  Q  cannot, as we had assumed, be embedded in a quandle  Conj(G). 

Remark:  A key property of  Q  that made the above reasoning possible is that the elements of each orbit of  Q  all induce the same permutation of  Q,  i.e., that  Q  satisfies the identity  (a ◁ b) ◁ c = b ◁ c.  Quandles satisfying this identity are studied in

  Victoria Lebed and Arnaud Mortier, Abelian quandles and quandles with abelian structure group, J. Pure Appl. Algebra, 225 (2021), no. 1, 106474, 22 pp., MR4116819, 

where they are called abelian quandles. 

I do not currently have plans of publishing the above example.