A386233 Number of good involutions of all nontrivial conjugation quandles of order A060652(n).

1, 32, 1, 17, 1, 13056, 66, 33, 1, 1

Offset: 1

Author

Luc Ta, Jul 16 2025

Keywords

• hard
• more
• nonn

Comments

A good involution f of a quandle Q is an involution that commutes with all inner automorphisms and satisfies the identity f(y)(x) = y^-1(x). We call the pair (Q,f) a symmetric quandle.

A conjugation quandle is a group viewed as a quandle under the conjugation operation. Since conjugation quandles of abelian groups are trivial, this sequence only considers nonabelian groups.

Examples

			For n = 1, 3, 5, 9, 10, there is a unique nonabelian group G of order A060652(n), and G is centerless. It follows from Ta, Prop. 5.3 that a(n) = 1.
		

References

• Seiichi Kamada, Quandles with good involutions, their homologies and knot invariants, Intelligence of Low Dimensional Topology 2006, World Scientific Publishing Co. Pte. Ltd., 2007, 101-108.

Links

• Lực Ta, Good involutions of conjugation subquandles, arXiv:2505.08090 [math.GT], 2025. See Table 2.
• Lực Ta, Symmetric-Rack-Classification, GitHub, 2025.

Crossrefs

Cf. A000001, A060652, A060689, A386233, A181769, A383828, A386231, A386232.

Other sequences related to racks and quandles: A383144, A181771, A176077, A179010, A193024, A254434, A177886, A196111, A226173, A236146, A248908, A165200, A242044, A226193, A242275, A243931, A257351, A198147, A225744, A226172, A226174, A383828-A383831, A383145, A383146, A178432, A385041, A383145, A181770.

Programs

GAP

See Ta, GitHub link