TY - JOUR

T1 - Automorphism groups of quandles arising from groups

AU - Bardakov, Valeriy G.

AU - Dey, Pinka

AU - Singh, Mahender

N1 - Publisher Copyright: © 2016, Springer-Verlag Wien.

PY - 2017/12/1

Y1 - 2017/12/1

N2 - 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.

AB - 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.

KW - Automorphism of quandle

KW - Central automorphism

KW - Connected quandle

KW - Knot quandle

KW - Two-point homogeneous quandle

KW - ALEXANDER QUANDLES

UR - http://www.scopus.com/inward/record.url?scp=84991401413&partnerID=8YFLogxK

U2 - 10.1007/s00605-016-0994-x

DO - 10.1007/s00605-016-0994-x

M3 - Article

AN - SCOPUS:84991401413

VL - 184

SP - 519

EP - 530

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 4

ER -