Automorphism groups of quandles arising from groups

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Automorphism groups of quandles arising from groups. / Bardakov, Valeriy G.; Dey, Pinka; Singh, Mahender.

In: Monatshefte fur Mathematik, Vol. 184, No. 4, 01.12.2017, p. 519-530.

Research output: Contribution to journal › Article › peer-review

Harvard

Bardakov, VG, Dey, P & Singh, M 2017, 'Automorphism groups of quandles arising from groups', Monatshefte fur Mathematik, vol. 184, no. 4, pp. 519-530. https://doi.org/10.1007/s00605-016-0994-x

APA

Bardakov, V. G., Dey, P., & Singh, M. (2017). Automorphism groups of quandles arising from groups. Monatshefte fur Mathematik, 184(4), 519-530. https://doi.org/10.1007/s00605-016-0994-x

Vancouver

Bardakov VG, Dey P, Singh M. Automorphism groups of quandles arising from groups. Monatshefte fur Mathematik. 2017 Dec 1;184(4):519-530. doi: 10.1007/s00605-016-0994-x

Author

Bardakov, Valeriy G. ; Dey, Pinka ; Singh, Mahender. / Automorphism groups of quandles arising from groups. In: Monatshefte fur Mathematik. 2017 ; Vol. 184, No. 4. pp. 519-530.

BibTeX

@article{e257de66b17246ebba19cc2e46006eea,

title = "Automorphism groups of quandles arising from groups",

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.",

keywords = "Automorphism of quandle, Central automorphism, Connected quandle, Knot quandle, Two-point homogeneous quandle, ALEXANDER QUANDLES",

author = "Bardakov, {Valeriy G.} and Pinka Dey and Mahender Singh",

note = "Publisher Copyright: {\textcopyright} 2016, Springer-Verlag Wien.",

year = "2017",

month = dec,

day = "1",

doi = "10.1007/s00605-016-0994-x",

language = "English",

volume = "184",

pages = "519--530",

journal = "Monatshefte fur Mathematik",

issn = "0026-9255",

publisher = "Springer-Verlag GmbH and Co. KG",

number = "4",

}

RIS

TY - JOUR

T1 - Automorphism groups of quandles arising from groups

AU - Bardakov, Valeriy G.

AU - Dey, Pinka

AU - Singh, Mahender

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 -

ID: 9866165