scholarly journals A Note on Counting Flows in Signed Graphs

10.37236/7958 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Matt DeVos ◽  
Edita Rollová ◽  
Robert Šámal
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Fundamental Theorem ◽  
Signed Graph ◽  
Signed Graphs ◽  
Odd Order ◽  
A Finite Group ◽  
Special Case

Tutte initiated  the study of nowhere-zero flows and proved the following fundamental theorem: For every graph $G$ there is a polynomial $f$ so that for every abelian group $\Gamma$ of order $n$, the number of nowhere-zero $\Gamma$-flows in $G$ is $f(n)$. For signed graphs (which have bidirected orientations), the situation is more subtle. For a finite group $\Gamma$, let $\epsilon_2(\Gamma)$ be the largest integer $d$ so that $\Gamma$ has a subgroup isomorphic to $\mathbb{Z}_2^d$. We prove that for every signed graph $G$ and $d \ge 0$ there is a polynomial $f_d$ so that $f_d(n)$ is the number of nowhere-zero $\Gamma$-flows in $G$ for every abelian group $\Gamma$ with $\epsilon_2(\Gamma) = d$ and $|\Gamma| = 2^d n$. Beck and Zaslavsky [JCTB 2006] had previously established the special case of this result when $d=0$ (i.e., when $\Gamma$ has odd order).

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

scholarly journals On Schur p-Groups of odd order

2017 ◽  
Vol 16 (03) ◽  
pp. 1750045 ◽  
Author(s):  
Grigory Ryabov
Keyword(s):  
Finite Group ◽  
Odd Order ◽  
A Finite Group ◽  
Natural Way

A finite group [Formula: see text] is called a Schur group if any [Formula: see text]-ring over [Formula: see text] is associated in a natural way with a subgroup of [Formula: see text] that contains all right translations. We prove that the groups [Formula: see text], where [Formula: see text], are Schur. Modulo previously obtained results, it follows that every noncyclic Schur [Formula: see text]-group, where [Formula: see text] is an odd prime, is isomorphic to [Formula: see text] or [Formula: see text], [Formula: see text].

scholarly journals Integral Cayley Graphs over Abelian Groups

10.37236/353 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Additive Group ◽  
Cyclic Groups ◽  
Vertex Set ◽  
A Finite Group ◽  
Gamma S

Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.

Generalized Casimir Operators

1969 ◽  
Vol 21 ◽  
pp. 1496-1505
Author(s):  
A. J. Douglas
Keyword(s):  
Finite Group ◽  
Casimir Operator ◽  
Identity Element ◽  
Ring Homomorphism ◽  
Fixed Ring ◽  
Injective Modules ◽  
Casimir Operators ◽  
Maschke’S Theorem ◽  
A Finite Group ◽  
Special Case

Throughout this paper, S will be a ring (not necessarily commutative) with an identity element ls ≠ 0s. We shall use R to denote a second ring, and ϕ: S→ R will be a fixed ring homomorphism for which ϕ1S = 1R.In (7), Higman generalized the Casimir operator of classical theory and used his generalization to characterize relatively projective and injective modules. As a special case, he obtained a theorem which contains results of Eckmann (3) and of Higman himself (5), and which also includes Gaschütz's generalization (4) of Maschke's theorem. (For a discussion of some of the developments of Maschke's idea of averaging over a finite group, we refer the reader to (2, Chapter IX).) In the present paper, we define the Casimir operator of a family of S-homomorphisms of one R-module into another, and we again use this operator to characterize relatively projective and injective modules.

Subgroups of Central Separable Algebras

1973 ◽  
Vol 25 (4) ◽  
pp. 881-887 ◽  
Author(s):  
E. D. Elgethun
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Division Ring ◽  
Multiplicative Group ◽  
Prime Field ◽  
Odd Order ◽  
Multiplicative Subgroup ◽  
A Finite Group ◽  
Separable Algebras

In [8] I. N. Herstein conjectured that all the finite odd order sub-groups of the multiplicative group in a division ring are cyclic. This conjecture was proved false in general by S. A. Amitsur in [1]. In his paper Amitsur classifies all finite groups which can appear as a multiplicative subgroup of a division ring. Let D be a division ring with prime field k and let G be a finite group isomorphic to a multiplicative subgroup of D.

The Inverse Multiplier for Abelian Group Difference Sets

1964 ◽  
Vol 16 ◽  
pp. 787-796 ◽  
Author(s):  
E. C. Johnsen
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Collineation Group ◽  
Difference Sets ◽  
Difference Set ◽  
The One ◽  
A Finite Group

In(1)Bruck introduced the notion of a difference set in a finite group. LetGbe a finite group ofvelements and let D = {di},i= 1, . . . ,kbe ak-subset ofGsuch that in the set of differences {di-1dj} each element ≠ 1 inGappears exactly λ times, where 0 < λ <k<v— 1. When this occurs we say that (G,D) is av,k,λ group difference set. Bruck showed that this situation is equivalent to the one where the differences {didj-1} are considered instead, and that av,k, λ group difference set is equivalent to a transitivev,k,λconfiguration, i.e., av,k,λconfiguration which has a collineation group which is transitive and regular on the elements (points) and on the blocks (lines) of the configuration. Among the parametersv,kandλ, then, we have the relation shown by Ryser(5)

scholarly journals Finite Groups Which Admit An Automorphism With Few Orbits

1960 ◽  
Vol 12 ◽  
pp. 73-100 ◽  
Author(s):  
Daniel Gorenstein
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Groups ◽  
Study Groups ◽  
Quaternion Group ◽  
Fixed Integer ◽  
Fixed Element ◽  
Odd Order ◽  
Special Case

In the course of investigating the structure of finite groups which have a representation in the form ABA, for suitable subgroups A and B, we have been forced to study groups G which admit an automorphism ϕ such that every element of G lies in at least one of the orbits under ϕ of the elements g, gϕr(g), gϕrϕ(g)ϕ2r(g), gϕr(g)ϕr2r(g)ϕ3r(g), etc., where g is a fixed element of G and r is a fixed integer.In a previous paper on ABA-groups written jointly with I. N. Herstein (4), we have treated the special case r = 0 (in which case every element of G can be expressed in the form ϕi(gj)), and have shown that if the orders of ϕ and g are relatively prime, then G is either Abelian or the direct product of an Abelian group of odd order and the quaternion group of order 8.

scholarly journals A connection between the number of subgroups and the order of a finite group

2020 ◽  
pp. 2250001
Author(s):  
Mihai-Silviu Lazorec
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Abelian Group ◽  
Subgroup Lattice ◽  
Minimum Value ◽  
A Finite Group

For a finite group [Formula: see text], we associate the quantity [Formula: see text], where [Formula: see text] is the subgroup lattice of [Formula: see text]. Different properties and problems related to this ratio are studied throughout this paper. We determine the second minimum value of [Formula: see text] on the class of [Formula: see text]-groups of order [Formula: see text], where [Formula: see text] is an integer. We show that the set containing the quantities [Formula: see text], where [Formula: see text] is a finite (abelian) group, is dense in [Formula: see text] Finally, we consider [Formula: see text] to be a function on [Formula: see text] and we indicate some of its properties, the main result being the classification of finite abelian [Formula: see text]-groups [Formula: see text] satisfying [Formula: see text]

scholarly journals Equivariant Map Queer Lie Superalgebras

2016 ◽  
Vol 68 (2) ◽  
pp. 258-279 ◽  
Author(s):  
Lucas Calixto ◽  
Adriano Moura ◽  
Alistair Savage
Keyword(s):  
Finite Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Equivariant Map ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Regular Maps ◽  
A Finite Group ◽  
Special Case

AbstractAn equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

scholarly journals Impartial achievement games for generating nilpotent groups

2019 ◽  
Vol 22 (3) ◽  
pp. 515-527
Author(s):  
Bret J. Benesh ◽  
Dana C. Ernst ◽  
Nándor Sieben
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Generating Set ◽  
Odd Order ◽  
Impartial Game ◽  
A Finite Group

AbstractWe study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form{T\times H}, whereTis a 2-group andHis a group of odd order. This includes all nilpotent and hence abelian groups.

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