The Experts below are selected from a list of 87 Experts worldwide ranked by ideXlab platform
Bridget S. Webb - One of the best experts on this subject based on the ideXlab platform.
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Small Partial Latin Squares that Cannot be Embedded in a Cayley Table
The Australasian Journal of Combinatorics, 2017Co-Authors: Ian M. Wanless, Bridget S. WebbAbstract:We answer a question posed by Denes and Keedwell that is equivalent to the following. For each order n what is the smallest size of a partial latin square that cannot be embedded into the Cayley Table of any group of order n? We also solve some variants of this question and in each case classify the smallest examples that cannot be embedded. We close with a question about embedding of diagonal partial latin squares in Cayley Tables.
Ian M. Wanless - One of the best experts on this subject based on the ideXlab platform.
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Small Partial Latin Squares that Cannot be Embedded in a Cayley Table
The Australasian Journal of Combinatorics, 2017Co-Authors: Ian M. Wanless, Bridget S. WebbAbstract:We answer a question posed by Denes and Keedwell that is equivalent to the following. For each order n what is the smallest size of a partial latin square that cannot be embedded into the Cayley Table of any group of order n? We also solve some variants of this question and in each case classify the smallest examples that cannot be embedded. We close with a question about embedding of diagonal partial latin squares in Cayley Tables.
Stephen P Humphries - One of the best experts on this subject based on the ideXlab platform.
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weak Cayley Table groups of some crystallographic groups
Glasgow Mathematical Journal, 2018Co-Authors: Stephen P Humphries, Rebeca A PaulsenAbstract:For a group G , a weak Cayley Table isomorphism is a bijection f : G → G such that f ( g 1 g 2 ) is conjugate to f ( g 1 ) f ( g 2 ) for all g 1 , g 2 ∈ G . The set of all weak Cayley Table isomorphisms forms a group ( G ) that is the group of symmetries of the weak Cayley Table of G . We determine ( G ) for each of the 17 wallpaper groups G , and for some other crystallographic groups.
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weak Cayley Table groups iii psl 2 q
Communications in Algebra, 2017Co-Authors: Stephen P Humphries, Long NguyenAbstract:ABSTRACTA weak Cayley Table isomorphism is a bijection φ:G→H of groups such that φ(xy)∼φ(x)φ(y) for all x,y∈G. Here ∼ denotes conjugacy. When G = H the set of all weak Cayley Table isomorphisms φ:G→G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I:G→G,x↦x−1. Let 𝒲0(G) = ⟨Aut(G),I⟩≤𝒲(G) and say that G has trivial weak Cayley Table group if 𝒲(G) = 𝒲0(G). We show that PSL(2,pn) has trivial weak Cayley Table group, where p≥5 is a prime and n≥1.
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weak Cayley Table groups of some crystallographic groups
arXiv: Group Theory, 2016Co-Authors: Stephen P Humphries, Rebeca A PaulsenAbstract:For a group $G$, a weak Cayley isomorphism is a bijection $f:G \to G$ such that $f(g_1g_2)$ is conjugate to $ f(g_1)f(g_2)$ for all $g_1,g_2 \in G$. They form a group $\mathcal W(G)$ that is the group of symmetries of the weak Cayley Table of $G$. We determine $\mathcal W(G)$ for each of the seventeen wallpaper groups $G$, and for some other crystallographic groups.
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weak Cayley Table groups ii alternating groups and finite coxeter groups
Communications in Algebra, 2015Co-Authors: Stephen P Humphries, Long NguyenAbstract:A weak Cayley Table isomorphism is a bijection φ: G → H of groups such that φ(xy) ∼ φ(x)φ(y) for all x, y ∈ G. Here ∼denotes conjugacy. When G = H the set of all weak Cayley Table isomorphisms φ: G → G forms a group 𝒲(G) that contains the automorphism group Aut(G) and the inverse map I: G → G, x → x −1. Let 𝒲0(G) = ⟨Aut(G), I⟩ ≤ 𝒲(G) and say that G has trivial weak Cayley Table group if 𝒲(G) = 𝒲0(G). We show that all finite irreducible Coxeter groups (except possibly E 8) have trivial weak Cayley Table group, as well as most alternating groups. We also consider some sporadic simple groups.
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weak Cayley Table groups
Journal of Algebra, 1999Co-Authors: Stephen P HumphriesAbstract:Abstract Let G , H be groups. We study bijections f : G → H which preserve conjugacy classes and satisfy: for all g , g ′ ∈ G the elements f ( gg ′) and f ( g ) f ( g ′) are conjugate. These are related to 2-characteristics of groups, objects originally studied by Frobenius and more recently by K. W. Johnson et. al. If G = H , then any automorphism or anti-automorphism satisfies these conditions. We call these trivial. We show, among other things, that any such f : G → G for G being the symmetric group S n , n > 0, dihedral group D n , n ∈ N ∪ {∞}, or the free group F n , n ≠ 3, is trivial.
Rebeca A Paulsen - One of the best experts on this subject based on the ideXlab platform.
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weak Cayley Table groups of some crystallographic groups
Glasgow Mathematical Journal, 2018Co-Authors: Stephen P Humphries, Rebeca A PaulsenAbstract:For a group G , a weak Cayley Table isomorphism is a bijection f : G → G such that f ( g 1 g 2 ) is conjugate to f ( g 1 ) f ( g 2 ) for all g 1 , g 2 ∈ G . The set of all weak Cayley Table isomorphisms forms a group ( G ) that is the group of symmetries of the weak Cayley Table of G . We determine ( G ) for each of the 17 wallpaper groups G , and for some other crystallographic groups.
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weak Cayley Table groups of some crystallographic groups
arXiv: Group Theory, 2016Co-Authors: Stephen P Humphries, Rebeca A PaulsenAbstract:For a group $G$, a weak Cayley isomorphism is a bijection $f:G \to G$ such that $f(g_1g_2)$ is conjugate to $ f(g_1)f(g_2)$ for all $g_1,g_2 \in G$. They form a group $\mathcal W(G)$ that is the group of symmetries of the weak Cayley Table of $G$. We determine $\mathcal W(G)$ for each of the seventeen wallpaper groups $G$, and for some other crystallographic groups.
Dmitri I Panyushev - One of the best experts on this subject based on the ideXlab platform.
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fredman s reciprocity invariants of abelian groups and the permanent of the Cayley Table
Journal of Algebraic Combinatorics, 2011Co-Authors: Dmitri I PanyushevAbstract:Let be the regular representation of a finite abelian group G and let ${\mathcal{C}}_{n}$ denote the cyclic group of order n. For $G={\mathcal{C}}_{n}$ , we compute the Poincare series of all ${\mathcal{C}}_{n}$ -isotypic components in (the symmetric tensor exterior algebra of ). From this we derive a general reciprocity and some number-theoretic identities. This generalises results of Fredman and Elashvili---Jibladze. Then we consider the Cayley Table, , of G and some generalisations of it. In particular, we prove that the number of formally different terms in the permanent of equals , where n is the order of G.
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fredman s reciprocity invariants of abelian groups and the permanent of the Cayley Table
arXiv: Representation Theory, 2010Co-Authors: Dmitri I PanyushevAbstract:Let $R$ be the regular representation of a finite abelian group $G$ and let $C_n$ denote the cyclic group of order $n$. For $G=C_n$, we compute the Poincare series of all $C_n$-isotypic components in $S^{\cdot} R\otimes \wedge^{\cdot} R$ (the symmetric tensor exterior algebra of $R$). From this we derive a general reciprocity and some number-theoretic identities. This generalises results of Fredman and Elashvili-Jibladze. Then we consider the Cayley Table, $M_G$, of $G$ and some generalisations of it. In particular, we prove that the number of formally different terms in the permanent of $M_G$ equals $(S^n R)^G$, where $n$ is the order of $G$.