Automorphisms and antiautomorphisms of quandles

In this paper we provide the conditions under which an automorphism or an antiautomorphism of a group $G$ induces an automorphism or an antiautomorphism of the $m$-conjugation quandle $\operatorname{Conj_{m}}(G),,, m\in \mathbb{Z} $, the core quandle $\operatorname{Core}(G)$, the generalized Alexander quandle $\operatorname{Alex}(G,ϕ)$ where $ϕ\in \operatorname{Aut}(G)$ and some others. We also construct automorphisms of these quandles that do not originate from $G$.

💡 Research Summary

The paper investigates how automorphisms and antiautomorphisms of a group G induce corresponding maps on several quandle constructions derived from G. The main families considered are the m‑conjugation quandles Conjₘ(G) (including the usual conjugation quandle when m=1), the core quandle Core(G), the generalized Alexander quandle Alex(G, ϕ) for a given group automorphism ϕ, and a collection of one‑parameter verbal quandles P_i (i=1,…,4) defined by a fixed element c∈G. The author systematically determines necessary and sufficient conditions for a group automorphism φ∈Aut(G) or a group antiautomorphism ψ∈AAut(G) to become an automorphism or antiautomorphism of the associated quandle.

Key results include:

1. Automorphisms of Conjₘ(G). The subgroup H consisting of left‑multiplications by central elements together with Aut(G) form a semidirect product H⋊Aut(G) that embeds in Aut(Conjₘ(G)) for any integer m (Proposition 2.1). When m=1, the outer automorphism group satisfies H⋊Out(G)≤Out(Conj(G)).

2. Antiautomorphisms of Conjₘ(G). An antiautomorphism ψ∈AAut(G) yields an automorphism of Conjₘ(G) iff every element y∈G satisfies y^{2m}∈Z(G) (Theorem 2.2(a)). For non‑trivial groups this condition rarely holds, and in fact no antiautomorphisms exist for Conjₘ(G) when G has trivial center (Corollary 2.3). Concrete examples with symmetric groups Σₙ and free groups illustrate the non‑existence.

3. Generalized Alexander Quandles. A group antiautomorphism ψ that commutes with ϕ (i.e., ψϕ=ϕψ) induces an automorphism of Alex(G,ϕ) iff ϕ is a central automorphism (Theorem 2.6(a)). The same ψ induces an antiautomorphism of Alex(G,ϕ) only when G is abelian (Theorem 2.6(b)). Consequently, the intersection C_{Aut(G)}(ϕ)∩AAut(Alex(G,ϕ)) is non‑empty precisely for abelian G (Theorem 2.7).

4. Core Quandles. Every group antiautomorphism is an automorphism of Core(G) (Theorem 3.1(a)). However, a group antiautomorphism is a quandle antiautomorphism of Core(G) iff G has exponent 3, i.e., (x^{-1}y)^3=1 for all x,y∈G (Theorem 3.1(b)). This leads to the observation that Core(G) is commutative exactly when G has exponent 3, and in that case AAut(Core(G))=Aut(Core(G)).

5. Dihedral Quandles. For the dihedral quandle Rₙ with n≠3, the antiautomorphism set is empty (Theorem 3.6), while the automorphism group is the well‑known semidirect product ℤₙ⋊ℤₙ^×.

6. Construction of New Automorphisms. Beyond those induced by Aut(G), the paper constructs automorphisms using the group F={f_{a,b}(x)=axb} and its normal subgroup H. In particular, G^{op}⋊C_{Aut(G)}(ϕ) embeds in Aut(Alex(G,ϕ)) (Proposition 2.8), showing that the opposite group structure contributes non‑trivial symmetries.

7. One‑Parameter Verbal Quandles P_i. For a fixed c∈G, the quandle P_i admits an automorphism induced by φ∈Aut(G) iff c^{-1}φ^{-1}(c)∈Z(G) (Theorem 5.1). If c∈Fix(φ), then φ induces a quandle antiautomorphism precisely when x=