## Abstract

Let G be a group and φ∈ Aut (G). Then the set G equipped with the binary operation a∗ b= φ(ab^{- 1}) b gives a quandle structure on G, denoted by Alex (G, φ) , and called the generalised Alexander quandle of G with respect to φ. When G is an additive abelian group and φ= - id _{G}, then Alex (G, φ) is the well-known Takasaki quandle of G. In this paper, we determine the group of automorphisms and inner automorphisms of Takasaki quandles of abelian groups with no 2-torsion, and Alexander quandles of finite abelian groups with respect to fixed-point free automorphisms. As an application, we prove that if G≅ (Z/ pZ) ^{n} and φ is multiplication by a non-trivial unit of Z/ pZ, then Aut (Alex (G, φ)) acts doubly transitively on Alex (G, φ). This generalises a recent result of Ferman et al. (J Knot Theory Ramifications 20:463–468, 2011) for quandles of prime order.

Original language | English |
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Pages (from-to) | 519-530 |

Number of pages | 12 |

Journal | Monatshefte fur Mathematik |

Volume | 184 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Dec 2017 |

## Keywords

- Automorphism of quandle
- Central automorphism
- Connected quandle
- Knot quandle
- Two-point homogeneous quandle
- ALEXANDER QUANDLES