Elsevier

Journal of Algebra

Volume 573, 1 May 2021, Pages 410-435
Journal of Algebra

Relations between quandle extensions and group extensions

https://doi.org/10.1016/j.jalgebra.2020.12.038Get rights and content

Abstract

In [6] and [7], Joyce and Matveev showed that for given a group G and an automorphism ϕ, there is a quandle structure on the underlying set of G. When the automorphism is an inner-automorphism by ζ, we denote this quandle structure as (G,ζ).

In this paper, we show a relationship between group extensions of a group G and quandle extensions of the quandle (G,ζ). In fact, there exists a group homomorphism from Hgp2(G;A) to Hq2((G,ζ);A). Next, we show a relationship between quandle extensions of a quandle Q and quandle extensions of the quandle on the inner automorphism group of Q. Indeed, there exists a group homomorphism from Hq2(Q;A) to Hq2((Inn(Q),ζ);A). Finally, we observe via examples a relationship between extensions of a quandle and extensions of the inner automorphism group of the quandle.

Section snippets

Preliminaries

A quandle is an algebraic structure consisting of a set Q and a binary operation :Q×QQ satisfying

  • (i)

    for any xQ,xx=x,

  • (ii)

    for any x,yQ, there uniquely exists zQ such that zx=y,

  • (iii)

    (xy)z=(xz)(yz) for all x,y,zQ.

Example 1.1

Examples of quandles include the following:

  • (1)

    for any set Q, we define a quandle structure by xy=x for all x,yQ. The quandle is called the trivial quandle;

  • (2)

    Q(4,1) is the quandle of order 4 with the quandle operation in Table 1, called the tetrahedral quandle. Indeed, the quandle operation

Group extension and quandle extension of a group

In [6] and [7], Joyce and Matveev showed that a group can be seen as a quandle as follows. Let G be a group. Let φ be a group automorphism on G. Define a binary operation on G by gφh=φ(gh1)h. Then G is a quandle under the operation φ. For a fixed element ζ in G, let φζ:GG denote conjugation by ζ, indeed, φζ(g)=ζ1gζ for gG. We denote the operation φζ by ζ simply.

Let 0AιG˜πG1 be a central group extension of G by an abelian group A. Let s:GG˜ be a section satisfying s(1)=(1,0) and s(ζ)

Quandle extension and group extension of quandle automorphism groups

Let Q be a quandle. Consider its automorphism group Aut(Q) as a quandle. In this section, we will show a relationship between extensions of a quandle Q and extensions of the quandle (Inn(Q),ζ). In fact, we will show that there exists a group homomorphism from Hq2(Q;A) to Hq2((Inn(Q),ζ);A). (Theorem 3.7, Theorem 3.9)

Let G be a group and φ a group automorphism on G. Then G is a quandle under the operation gφh=φ(gh1)h, as seen in section 2. Let H be the subgroup of G whose elements are fixed

Quandle extension of a quandle and group extension of its inner automorphism group

In this section, we observe a relationship between abelian extensions of a quandle and central extensions of the inner automorphism group of the quandle with some examples.

Example 4.1

Consider abelian extensions of Q(4,1) in Example 1.9. As mentioned before, the resulting non-trivial abelian extension of Q(4,1) by Z2 is isomorphic to Q(8,1) and the quandle projection p:Q(8,1)Q(4,1) is given by {0,4}0, {1,5}1, {2,6}2, {3,7}3.

Recall that the inner automorphism group of Q(4,1) is the alternating group A4

CRediT authorship contribution statement

Yongju Bae: Conceptualization, Methodology, J. Scott Carter: Supervision, Writing-Reviewing and Editing, Byeorhi Kim: Calculation, Writing-Original draft preparation.

References (9)

  • Y. Bae et al.

    Amusing permutation representations of group extensions

  • J.S. Carter et al.

    Extensions of quandles and cocycle knot invariants

    J. Knot Theory Ramif.

    (2003)
  • J.S. Carter et al.

    Quandle cohomology and state-sum invariants of knotted curves and surfaces

    Trans. Am. Math. Soc.

    (2003)
  • J.S. Carter et al.

    Diagrammatic computations for quandles and cocycle knot invariants

There are more references available in the full text version of this article.

Cited by (0)

1

Yongju Bae was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2016R1D1A3B01007669).

2

J. Scott Carter was supported by the Japan Society for the Promotion of Science (L-18511) August 1, 2019 through May 1, 2020, and ICERM Sept 2020 through Dec 7, 2020.

3

Byeorhi Kim was partially supported by National Research Foundation of Korea (NRF) Grant No. 2019R1A3B2067839.

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