Connected Quandles with Order Equal to Twice an Odd Prime

McCarron, James P.

doi:10.48550/arxiv.1210.2150

Cited by 4 publications

(5 citation statements)

References 13 publications

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“…The following notion of 2-transitivity (and also higher k-transitivity) of quandles was introduced in [17,18]. Two-transitive quandles are called two-point homogeneous quandles in [27].…”

Section: 2mentioning

confidence: 99%

Ring theoretic aspects of quandles

2019

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We associate to every quandle X and an associative ring with unity k, a nonassociative ring k[X] following [3]. The basic properties of such rings are investigated. In particular, under the assumption that the inner automorphism group Inn(X) acts orbit 2transitively on X, a complete description of right (or left) ideals is provided. The complete description of right ideals for the dihedral quandles R n is given. It is also shown that if for two quandles X and Y the inner automorphism groups act 2-transitively and k[X] is isomorphic to k[Y], then the quandles are of the same partition type. However, we provide examples when the quandle rings k[X] and k[Y] are isomorphic, but the quandles X and Y are not isomorphic. These examples answer some open problems in [3].

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“…The following notion of 2-transitivity (and also higher k-transitivity) of quandles was introduced in [17,18]. Two-transitive quandles are called two-point homogeneous quandles in [27].…”

Section: 2mentioning

confidence: 99%

Ring theoretic aspects of quandles

2019

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“…This question has a negative answer. In [33] it was conjectured and later proved in [22] and then in [15] that there is no indecomposable quandle of size 2p for p > 5. In particular, this implies that the lattice with one top and one bottom element and an antichain of 2p incomparable elements in between is not the subrack lattice of a quandle.…”

Section: Questions and Problemsmentioning

confidence: 99%

“…Rack morphisms can be used to define and classify (finite) simple racks. Finite racks of small size were classified in [33,22,15].…”

Section: Introductionmentioning

confidence: 99%

On the lattice of subracks of the rack of a finite group

Heckenberger

Shareshian

Welker

2019

Trans. Amer. Math. Soc.

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In this paper we initiate the study of racks from the combined perspective of combinatorics and finite group theory. A rack R is a set with a self-distributive binary operation. We study the combinatorics of the partially ordered set R(R) of all subracks of R with inclusion as the order relation. Groups G with the conjugation operation provide an important class of racks. For the case R = G we show that• the order complex of R(R) has the homotopy type of a sphere,• the isomorphism type of R(R) determines if G is abelian, nilpotent, supersolvable, solvable or simple,In addition, we provide some examples of subracks R of a group G for which R(R) relates to well studied combinatorial structures. In particular, the examples show that the order complex of R(R) for general R is more complicated than in the case R = G. arXiv:1512.01459v1 [math.CO] 4 Dec 2015

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“…We shall also need the following lemma of [14]. Recall that a quandle is simple if it has no quotients except itself and the trivial quandle of one element [11].…”

Section: Lemma 7 (Mccarron)mentioning

confidence: 99%