scholarly journals Fredman’s reciprocity, invariants of abelian groups, and the permanent of the Cayley table

2010 ◽  
Vol 33 (1) ◽  
pp. 111-125
Author(s):  
Dmitri I. Panyushev
Keyword(s):  
Abelian Groups ◽  
Cayley Table

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.

scholarly journals Infinite Latin Squares: Neighbor Balance and Orthogonality

10.37236/8020 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Anthony B. Evans ◽  
Gage N. Martin ◽  
Kaethe Minden ◽  
M. A. Ollis
Keyword(s):  
Abelian Group ◽  
Infinite Order ◽  
Abelian Groups ◽  
Latin Squares ◽  
Infinite Group ◽  
Cayley Table ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Complete Mapping ◽  
Infinite Abelian Group

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 possible to permute the rows and columns of the Cayley table to create an infinite Vatican square. We also construct a Vatican square of any given infinite order that is not obtainable by permuting the rows and columns of a Cayley table.  Regarding orthogonality, we show that every infinite group $G$ has a set of $|G|$ mutually orthogonal orthomorphisms and hence there is a set of $|G|$ mutually orthogonal Latin squares based on $G$. We show that an infinite group $G$ with $|G|$-many square elements has a strong complete mapping; and, with some possible exceptions, infinite abelian groups have a strong complete mapping.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

Finite non-Abelian groups with complemented non-Abelian subgroups

1978 ◽  
Vol 29 (6) ◽  
pp. 541-544 ◽  
Author(s):  
P. P. Baryshovets
Keyword(s):  
Abelian Groups ◽  
Abelian Subgroups

scholarly journals Computing the number of symmetric colorings of elementary Abelian groups

2021 ◽  
Vol 60 (2) ◽  
pp. 2075-2081
Author(s):  
Yuliya Zelenyuk
Keyword(s):  
Abelian Groups

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

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