# Automorphism groups of quandles and related groups

## TL;DR

This paper investigates automorphism groups of quandles, providing conditions for their structure, classifying certain automorphism groups, and analyzing automorphisms of specific quandle extensions.

## Contribution

It characterizes automorphism groups of conjugation quandles, identifies when they are equal to the full automorphism group, and classifies finite quandles with highly transitive automorphism actions.

## Key findings

- Automorphism group of conjugation quandle equals Z(G) semidirect Aut(G) under specific conditions.
- Inner automorphism groups of Takasaki quandles are Coxeter groups.
- Finite quandles with 3 or more transitive automorphism group actions are classified.

## Abstract

In this paper we study different questions concerning automorphisms of quandles. For a conjugation quandle $Q={\rm Conj}(G)$ of a group $G$ we determine several subgroups of ${\rm Aut}(Q)$ and find necessary and sufficient conditions when these subgroups coincide with the whole group ${\rm Aut}(Q)$. In particular, we prove that ${\rm Aut}({\rm Conj}(G))={\rm Z}(G)\rtimes {\rm Aut}(G)$ if and only if either ${\rm Z}(G)=1$ or $G$ is one of the groups $\mathbb{Z}_2$, $\mathbb{Z}_2^2$ or $\mathbb{Z}_3$. For a big list of Takasaki quandles $T(G)$ of an abelian group $G$ with $2$-torsion we prove that the group of inner automorphisms ${\rm Inn}(T(G))$ is a Coxeter group. We study automorphisms of certain extensions of quandles and determine some interesting subgroups of the automorphism groups of these quandles. Also we classify finite quandles $Q$ with $3\leq k$-transitive action of ${\rm Aut}(Q)$.