Related papers: Magic Cayley-Sudoku Tables
A Cayley-Sudoku table of a finite group G is a Cayley table for G subdivided into uniformly sized rectangular blocks, in such a way that each group element appears once in each block. They were introduced by J. Carmichael, K. Schloeman, and…
A symmetry group for Sudoku is complete if its action partitions the set of Sudoku boards into all possible orbits, and minimal if no group of smaller size would do the same. Previously, for a 4 x 4 Sudoku variation known as Shidoku, the…
Based on the rules of magic cubes, a game of two-dimensional magic cube was deliberately designed. This essay will explore its properties with the assistance of group theory and computer programming. It will first elaborate the rules of…
We say that a group G is a cube group if it is generated by a set S of involutions such that the corresponding Cayley graph Cay(G,S) is isomorphic to a cube. Equivalently, G is a cube group if it acts on a cube such that the action is…
The mathematical aspects of the popular logic game Sudoku incorporate a significant number of the group theory concepts. In this note, we describe all symmetric transformations of the Sudoku grid. We do not intend to obtain a new strategy…
A connected linear algebraic group G is called a Cayley group if the Lie algebra of G endowed with the adjoint G-action and the group variety of G endowed with the conjugation G-action are birationally G-isomorphic. In particular, the…
Calculations of the number of equivalence classes of Sudoku boards has to this point been done only with the aid of a computer, in part because of the unnecessarily large symmetry group used to form the classes. In particular, the…
We present systematic methods of constructing pandiagonal sudoku squares of order k*k and Knut Vik sudoku squares of order k*k not divisible by 2 or 3. Pandiagonal magic squares are constructed from these squares. Examples of all these…
Regarding neighbor balance, we consider natural generalizations of $D$-complete Latin squares and Vatican squares from the finite to the infinite. We show that if $G$ is an infinite abelian group with $|G|$-many square elements, then it is…
A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e., a G-equivariant birational isomorphism over k between the group variety G and the Lie algebra Lie(G). A Cayley map can be thought of as a…
It is a fun game to complete a partial character table of a finite group. We show that one can reconstruct a missing row or column from a given table. The proof relies on deep properties of fully ramified characters. Moreover, we extend a…
We show that every finitely generated group G with an element of order at least $(5rank(G))^{12}$ admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above…
Group theory involves the study of symmetry, and its inherent beauty gives it the potential to be one of the most accessible and enjoyable areas of mathematics, for students and non-mathematicians alike. Unfortunately, many students never…
Let $\Gamma$ be a group of order $n^2$ and $SMS_{\Gamma}(n)=(a_{i,j})_{n\times n}$ be an $n\times n$ array whose entries are all distinct elements of $\Gamma$. If there exists an element $\mu\in\Gamma$ such that for every row $i$, there…
In this paper we study the rotation and spatial inversion symmetry of regular tetrahedron. We obtain the representation matrix, multiplication table,the order of all group elements, all possible combinations of generator elements, the…
A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a…
A perfect code $C$ in a graph $\Gamma$ is an independent set of vertices of $\Gamma$ such that every vertex outside of $C$ is adjacent to a unique vertex in $C$, and a total perfect code $C$ in $\Gamma$ is a set of vertices of $\Gamma$ such…
An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. The group ring $RG$ of a finite group $G$ is isomorphic to the set of {\em group ring matrices} over $R$. It is shown that…
A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e. a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra Lie(G). A prototypical example is the classical…
A quandle is an algebraic structure whose axioms correspond to the Reidemeister moves of knot theory. S. Kamada introduced the notion of a quandle with a good involution, which is later called a symmetric quandle. We are interested in the…