Related papers: Doubly transitive groups and cyclic quandles

In the paper we describe the class of principal quandles and show that connected quandles can be decomposed as a disjoint union of principal quandles. We also prove that simple affine quandles are finite and they can be characterized among…

Quandles are self-distributive algebraic structures known as sources of strong knots invariants, but also appearing in other contexts. From any quandle, one can construct two invariants: the structure group and the second quandle homology…

We prove that an Alexander quandle of prime order is generated by any pair of distinct elements. Furthermore, we prove for such a quandle that any ordered pair of distinct elements can be sent to any other such pair by an automorphism of…

A quandle of cyclic type of order $n$ with $f\geq 2$ fixed points is such that each of its permutations splits into $f$ cycles of length $1$ and one cycle of length $n-f$. In this article we prove that there is only one such connected…

A positive integer $n$ is defined to be cyclic if and only if every group of size $n$ is cyclic. Equivalently, $n$ is cyclic if and only if $n$ is relatively prime to the number of positive integers less than $n$ that are relatively prime…

In this paper, we study quandles of cyclic type, which form a particular subclass of finite quandles. The main result of this paper describes the set of isomorphism classes of quandles of cyclic type in terms of certain cyclic permutations.…

We prove a conjecture of Dukes and Herke concerning the possible orders of a basis for the cyclic group Z_n, namely : For each k \in N there exists a constant c_k > 0 such that, for all n \in N, if A \subseteq Z_n is a basis of order…

The notion of formal duality in finite Abelian groups appeared recently in relation to spherical designs, tight sphere packings, and energy minimizing configurations in Euclidean spaces. For finite cyclic groups it is conjectured that there…

Carter, Jelsovsky, Kamada, Langford and Saito have defined an invariant of classical links associated to each element of the second cohomology of a finite quandle. We study these invariants for Alexander quandles of the form Z[t,t^{-1}]/(p,…

K. Harada conjectured for any finite group $G$, the product of sizes of all conjugacy classes is divisible by the product of degrees of all irreducible characters. We study this conjecture when $G$ is the general linear group over a finite…

Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite…

A rack is a set with a binary operation such that left multiplications are automorphisms of the set and a quandle is a rack satisfying a certain condition. For a finite connected rack the cycle type of the permutation defined by left…

A (left) quandle is connected if its left multiplication group acts transitively. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to so-called constant quandle cocycles that form a subset of quandle cocycles. A…

We present methods of constructing examples of quandles of order 3n, where n is greater or equal to 3. The necessary and sufficient conditions for the constructed examples to be (i) connected (ii) group (conjugate) (iii) involutory and (iv)…

An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k \geq 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if…

We establish a canonical correspondence between connected quandles and certain configurations in transitive groups, called quandle envelopes. This correspondence allows us to efficiently enumerate connected quandles of small orders, and…

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Laba and Meyerowitz, but the spectral-tile direction is far from being well-understood. The product of at most…

It is conjectured that for every pair $(\ell,m)$ of odd integers greater than 2 with $m \equiv 1\; \pmod{\ell}$, there exists a cyclic two-factorization of $K_{\ell m}$ having exactly $(m-1)/2$ factors of type $\ell^m$ and all the others of…

We show that the spectral set conjecture by Fuglede holds in the setting of cyclic groups of order $p^n q$, where $p$, $q$ are distinct primes and $n\geq1$. This means that a subset $E$ of such a group $G$ tiles the group by translation…

The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of…