Relations between quandle extensions and group extensions
Section snippets
Preliminaries
A quandle is an algebraic structure consisting of a set Q and a binary operation satisfying
- (i)
for any ,
- (ii)
for any , there uniquely exists such that ,
- (iii)
for all .
Example 1.1 Examples of quandles include the following: for any set Q, we define a quandle structure by for all . The quandle is called the trivial quandle; 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 . Then G is a quandle under the operation . For a fixed element ζ in G, let denote conjugation by ζ, indeed, for . We denote the operation by simply.
Let be a central group extension of G by an abelian group A. Let be a section satisfying and
Quandle extension and group extension of quandle automorphism groups
Let Q be a quandle. Consider its automorphism group as a quandle. In this section, we will show a relationship between extensions of a quandle Q and extensions of the quandle . In fact, we will show that there exists a group homomorphism from to . (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 , 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 in Example 1.9. As mentioned before, the resulting non-trivial abelian extension of by is isomorphic to and the quandle projection is given by , , , . Recall that the inner automorphism group of is the alternating group
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)
- et al.
Amusing permutation representations of group extensions
- et al.
Extensions of quandles and cocycle knot invariants
J. Knot Theory Ramif.
(2003) - et al.
Quandle cohomology and state-sum invariants of knotted curves and surfaces
Trans. Am. Math. Soc.
(2003) - et al.
Diagrammatic computations for quandles and cocycle knot invariants
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.