SINIF CEBİRİ YAKLAŞIMI ALTINDA 𝑫𝟐𝒅 ve 𝑪𝟑𝒊 NOKTA GRUPLARI

In this study, the point groups 𝐷2𝑑 and 𝐶3𝑖 which belong to tetragonal and trigonal crystal systems, respectively, are handled under the class sum approach. Symmetry groups were formed with symmetry elements that left these point groups unchanged and Cayley tables of related groups were obtained. Using these tables, the conjugates of the elements and the classes of the group were formed. Secular equations are written for each class sum obtained by the sum of the elements that make up the class. By solving these secular equations, the character vectors are obtained. Thus, the character tables were reconstructed with the calculated characters for both point groups under the class sum approach.

Completing Partial Transversals of Cayley Tables of Abelian Groups

In 2003 Grüttmüller proved that if $n\geqslant 3$ is odd, then a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $2$ is completable to a transversal. Additionally, he conjectured that a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $k$ is completable to a transversal if and only if $n$ is odd and either $n \in \{k, k + 1\}$ or $n \geqslant 3k - 1$. Cavenagh, Hämäläinen, and Nelson (in 2009) showed the conjecture is true when $k = 3$ and $n$ is prime. In this paper, we prove Grüttmüller’s conjecture for $k = 2$ and $k = 3$ by establishing a more general result for Cayley tables of Abelian groups of odd order.

Diagonal groups and arcs over groups

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $$m\ge 2$$ m ≥ 2 , a set of $$m+1$$ m + 1 partitions of a set $$\Omega $$ Ω , any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if $$m=2$$ m = 2 ), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have $$m+r$$ m + r partitions with $$r\ge 2$$ r ≥ 2 , any m of which are minimal elements of a Cartesian lattice. If $$m=2$$ m = 2 , this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For $$m>2$$ m > 2 , things are more restricted. Any $$m+1$$ m + 1 of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality $$m+r$$ m + r in the $$(m-1)$$ ( m - 1 ) -dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques.

FRACTAL MAGMAS AND PUBLIC-KEY CRYPTOGRAPHY

In this paper, we deal with magmas the simplest algebras with a single binary operation. The main result of our research is algorithms for generating chain of finite magmas based on the self-similarity principle of its Cayley tables. In this way the cardinality of a magmas domain is twice as large as the previous one for each magma in the chain, and its Cayley table has a block-like structure. As an example, we consider a cyclic semigroup of binary operations generated by a finite magmas operation with a low-cardinality domain, and a modify the Diffie-Hellman-Merkle key exchange protocol for this case.

Algebras of Binary Isolating Formulas for Theories of Root Products of Graphs

Algebras of distributions of binary isolating and semi-isolating formulas are derived objects for given theory and reflect binary formula relations between realizations of 1-types. These algebras are associated with the following natural classification questions: 1) for a given class of theories, determine which algebras correspond to the theories from this class and classify these algebras; 2) to classify theories from a given class depending on the algebras defined by these theories of isolating and semi-isolating formulas. Here the description of a finite algebra of binary isolating formulas unambiguously entails a description of the algebra of binary semi-isolating formulas, which makes it possible to track the behavior of all binary formula relations of a given theory. The paper describes algebras of binary formulae for root products. The Cayley tables are given for the obtained algebras. Based on these tables, theorems describing all algebras of binary formulae distributions for the root multiplication theory of regular polygons on an edge are formulated. It is shown that they are completely described by two algebras.

The Chromatic Number of Finite Group Cayley Tables

The chromatic number of a latin square $L$, denoted $\chi(L)$, is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies $\chi(L) \leq |L|+2$. If true, this would resolve a longstanding conjecture—commonly attributed to Brualdi—that every latin square has a partial transversal of size $|L|-1$. Restricting our attention to Cayley tables of finite groups, we prove two results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group $G$ has chromatic number $|G|$ or $|G|+2$, with the latter case occurring if and only if $G$ has nontrivial cyclic Sylow 2-subgroups. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For $|G|\geq 3$, this improves the best-known general upper bound from $2|G|$ to $\frac{3}{2}|G|$, while yielding an even stronger result in infinitely many cases.

Weak Cayley tables and generalized centralizer rings of finite groups

For a finite group G we study certain rings (k)G called k-S-rings, one for each k ≥ 1, where (1)G is the centraliser ring Z(ℂG) of G. These rings have the property that (k+1)G determines (k)G for all k ≥ 1. We study the relationship of (2)G with the weak Cayley table of G. We show that (2)G and the weak Cayley table together determine the sizes of the derived factors of G (noting that a result of Mattarei shows that (1)G = Z(ℂG) does not). We also show that (4)G determines G for any group G with finite conjugacy classes, thus giving an answer to a question of Brauer. We give a criteria for two groups to have the same 2-S-ring and a result guaranteeing that two groups have the same weak Cayley table. Using these results we find a pair of groups of order 512 that have the same weak Cayley table, are a Brauer pair, and have the same 2-S-ring.