The 2-Characters of a Group and the Weak Cayley Table

2019 ◽  
pp. 177-209
Author(s):  
Kenneth W. Johnson
Keyword(s):  
Cayley Table

scholarly journals The Monoid Consisting of Kuratowski Operations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Szymon Plewik ◽  
Marta Walczyńska
Keyword(s):  
Systematic Study ◽  
Cayley Table ◽  
Paper And Pencil

The paper fills gaps in knowledge about Kuratowski operations which are already in the literature. The Cayley table for these operations has been drawn up. Techniques, using only paper and pencil, to point out all semigroups and its isomorphism types are applied. Some results apply only to topology, and one cannot bring them out, using only properties of the complement and a closure-like operation. The arguments are by systematic study of possibilities.

scholarly journals The Chromatic Number of Finite Group Cayley Tables

10.37236/7874 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Luis Goddyn ◽  
Kevin Halasz ◽  
E. S. Mahmoodian
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Chromatic Number ◽  
Finite Groups ◽  
Upper Bound ◽  
Latin Square ◽  
Finite Abelian Groups ◽  
Cayley Table ◽  
Minimum Number ◽  
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.

scholarly journals Transversals, Plexes, and Multiplexes in Iterated Quasigroups

10.37236/7304 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Anna Taranenko
Keyword(s):  
Latin Hypercube ◽  
Cayley Table ◽  
Recurrence Formulas

A $d$-ary quasigroup of order $n$ is a $d$-ary operation over a set of cardinality $n$ such that the Cayley table of the operation is a $d$-dimensional latin hypercube of the same order. Given a binary quasigroup $G$, the $d$-iterated quasigroup $G^{\left[d\right]}$ is a $d$-ary quasigroup that is a $d$-time composition of $G$ with itself. A $k$-multiplex (a $k$-plex) $K$ in a $d$-dimensional latin hypercube $Q$ of order $n$ or in the corresponding $d$-ary quasigroup is a multiset (a set) of $kn$ entries such that each hyperplane and each symbol of $Q$ is covered by exactly $k$ elements of $K$. It is common to call 1-plexes transversals. In this paper we prove that there exists a constant $c(G,k)$ such that if a $d$-iterated quasigroup $G$ of order $n$ has a $k$-multiplex then for large $d$ the number of its $k$-multiplexes is asymptotically equal to $c(G,k) \left(\frac{(kn)!}{k!^n}\right)^{d-1}$. As a corollary we obtain that if the number of transversals in the Cayley table of a $d$-iterated quasigroup $G$ of order $n$ is nonzero then asymptotically it is $c(G,1)  n!^{d-1}$.  In addition, we  provide limit constants and recurrence formulas for the numbers of transversals in two iterated quasigroups of order $5$, characterize a typical $k$-multiplex and estimate numbers of partial $k$-multiplexes and transversals in $d$-iterated quasigroups. 

scholarly journals Transversals, Near Transversals, and Diagonals in Iterated Groups and Quasigroups

10.37236/9699 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Anna A. Taranenko
Keyword(s):  
Commutator Subgroup ◽  
Latin Hypercube ◽  
Cayley Table

Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as to coincide with those in the corresponding latin hypercube. We prove that if a group $G$ of order $n$ satisfies the Hall–Paige condition, then the number of transversals in $G[d]$ is equal to $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d}  (1 + o(1))$ for large $d$, where $G'$ is the commutator subgroup of $G$. For a general quasigroup $G$, we obtain similar estimations on the numbers of transversals and near transversals in $G[d]$  and develop a method for counting diagonals of other types in iterated quasigroups.

Weak Cayley table groups III:PSL(2,q)

2016 ◽  
Vol 45 (7) ◽  
pp. 3110-3136 ◽  
Author(s):  
Stephen P. Humphries ◽  
Long Nguyen
Keyword(s):  
Cayley Table

Weak Cayley tables and generalized centralizer rings of finite groups

2012 ◽  
Vol 153 (2) ◽  
pp. 281-318 ◽  
Author(s):  
STEPHEN P. HUMPHRIES ◽  
EMMA L. RODE
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Cayley Table ◽  
A Finite Group ◽  
Finite Conjugacy ◽  
Relationship Of ◽  
Cayley Tables ◽  
The Relationship

AbstractFor 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.

scholarly journals Geometrical Representation of Automata over Some Abelian Groups

2014 ◽  
Vol 8 (1) ◽  
Author(s):  
Y.S. Gan ◽  
W.H. Fong ◽  
N.H. Sarmin ◽  
S. Turaev
Keyword(s):  
Abelian Groups ◽  
Finite Automata ◽  
Cayley Table ◽  
Geometrical Representation

One of the classic models of automata is finite automata, which determine whether a string belongs to a particular language or not. The string accepted by automata is said to be recognized by that automata. Another type of automata, so-called Watson-Crick automata, with two reading heads that work on double-stranded tapes using the complimentary relation. Finite automata over groups extend the possibilities of finite automata and allow studying the properties of groups using finite automata. In this paper, we consider finite automata over some Abelian groups ℤn and ℤn × ℤn. The relation of Cayley table to finite automata diagram is introduced in the paper. Some properties of groups ℤn and ℤn × ℤn in terms of automata are also presented in this paper.

scholarly journals Completing Partial Transversals of Cayley Tables of Abelian Groups

10.37236/9386 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Jaromy Kuhl ◽  
Donald McGinn ◽  
Michael William Schroeder
Keyword(s):  
General Result ◽  
Abelian Groups ◽  
Cayley Table ◽  
Odd Order ◽  
Cayley Tables

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.

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