bibkey: ta2025goodinvolutions authors: Lực Ta year: 2025 title: Good involutions of conjugation subquandles doi: 10.48550/arXiv.2505.08090 url: https://arxiv.org/abs/2505.08090v5 claim: “Problem 11.7 asks whether the Corollary 6.12 upper bound on good involutions is always strict when |Z(H)| and |X/H| are at least two.” strata_touched:

• D5/S0/Certificates/Groups/TaGoodInvolutionBoundRefutation license: citation-only triage: anchor

Good involutions of conjugation subquandles

Lực Ta, arXiv:2505.08090v5 [math.GT], submitted 2025-05-12, version 5 dated 2025-08-04. The references below use printed page numbers.

Example 2.6, page 5:

Let G be a group, and define s : G → S_G by sending each element g ∈ G to the conjugation map defined by s_g(h) := ghg⁻¹ for all h ∈ G. Then Conj G := (G, s) is a quandle called a conjugation quandle or conjugacy quandle. Note that s_g⁻¹ = s_{g⁻¹} for all g ∈ G.

Lemma 2.10, page 5:

For all subsets X ⊆ G, the pair Conj X := (X, s|_X) is a subquandle of Conj G if and only if X is closed under conjugation by elements of the subgroup ⟨X⟩ of G generated by X.

Definition 2.17, page 6:

Given two racks (X, s) and (Y, t), we say that a map φ : X → Y is a rack homomorphism if φs_x = t_{φ(x)}φ for all x ∈ X. A rack isomorphism is a bijective rack homomorphism, and a rack automorphism of a rack R is a rack isomorphism from R to itself.

Definition 2.20, page 6:

We denote the automorphism group of a rack R = (X, s) by Aut R.

Definition 3.8, page 7:

Let R = (X, s) be a rack. A good involution of R is an involution ρ ∈ S_X that satisfies the equalities ρ s_x = s_x ρ, s_{ρ(x)} = s_x⁻¹ for all x ∈ X. We denote the set of good involutions of R by Good R.

Section 6.3.1, page 14:

In the following, let X/H denote the set of orbits of X under the action of H by conjugation.

Corollary 6.12, page 14:

Let k(X) := |X/H|. Then |Good(Conj X)| ≤ min(|Aut(Conj X)|, |Z(H)|^{k(X)}). If X is not closed under inverses, then the bound |Good(Conj X)| < |Z(H)|^{k(X)} is strict.

Problem 11.7, page 27:

In the setting of Corollary 6.12, suppose that |Z(H)|, |X/H| ≥ 2 (so, in particular, Conj X is not connected). Is the upper bound on |Good(Conj X)| in Corollary 6.12 always strict?

Here H = ⟨X⟩. The formal strictness claim quantifies over finite groups and finite conjugation subquandles, with the source definitions above. The subquandle X = {±i, ±j} of Q₈ has H = Q₈, |Z(H)| = 2, |X/H| = 2, |Good(Conj X)| = 4 and |Aut(Conj X)| = 8. Thus the upper bound is attained. The question does not require finite G; this finite counterexample also answers its unrestricted formulation. It makes no claim about strict bounds for any other class of subquandles.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2505.08090
• URL: https://arxiv.org/abs/2505.08090v5
• Text: https://arxiv.org/pdf/2505.08090v5, printed pages 5–7, 14 and 27: Example 2.6, Lemma 2.10, Definitions 2.17, 2.20 and 3.8, section 6.3.1, Corollary 6.12 and Problem 11.7. Source retrieved 2026-09-30.