Automorphism groups of quandles arising from groups

Valeriy G. Bardakov, Pinka Dey, Mahender Singh

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

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 languageEnglish
Pages (from-to)519-530
Number of pages12
JournalMonatshefte fur Mathematik
Volume184
Issue number4
DOIs
Publication statusPublished - 1 Dec 2017

Keywords

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

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