The Coherence Number of 2-Groups

1990 ◽  
Vol 33 (4) ◽  
pp. 503-508 ◽  
Author(s):  
James McCool
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Dihedral Group ◽  
Upper Bounds ◽  
Distinguished Element ◽  
A Finite Group

AbstractLet G be a finite group. A natural invariant c(G) of G has been defined by W.J. Ralph, as the order (possibly infinite) of a distinguished element of a certain abelian group associated to G. Ralph has shown that c(Zn) = 1 and c(Z2 ⴲ Z2) = 2. In the present paper we show that c(G) is finite whenever G is a dihedral group or a 2-group, and obtain upper bounds for c(G) in these cases.

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.

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 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]

Conjugate p-Subgroups of Finite Groups

1972 ◽  
Vol 24 (1) ◽  
pp. 17-28
Author(s):  
John J. Currano
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Finite Groups ◽  
Nilpotence Class ◽  
A Finite Group

Throughout this paper, let p be a prime, P be a p-group of order pt , and ϕ be an isomorphism of a subgroup R of P of index p onto a subgroup Q which fixes no non-identity subgroup of P, setwise. In [2, Lemma 2.2], Glauberman shows that P can be embedded in a finite group G such that ϕ is effected by conjugation by some element g of G. We assume that P is thus embedded. Then Q = P ∩ Pg. Let H = 〈P,Pg〉 and V = [H,Z(Q)], so Q ⊲ H and V ⊲ H.Let E(p) be the non-abelian group of order p3 which is generated by two elements of order p. Then E(p) is dihedral if p = 2 and has exponent p if p is odd. If p is odd, then E* (p) is defined in § 2 to be a particular group of order p6 and nilpotence class three.

scholarly journals On the group ring of a finite abelian group

1969 ◽  
Vol 1 (2) ◽  
pp. 245-261 ◽  
Author(s):  
Raymond G. Ayoub ◽  
Christine Ayoub
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Group Ring ◽  
Free Abelian Group ◽  
Finite Abelian Group ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Group Of Units ◽  
Direct Sum ◽  
A Finite Group

The group ring of a finite abelian group G over the field of rational numbers Q and over the rational integers Z is studied. A new proof of the fact that the group ring QG is a direct sum of cyclotomic fields is given – without use of the Maschke and Wedderburn theorems; it is shown that the projections of QG onto these fields are determined by the inequivalent characters of G. It is proved that the group of units of ZG is a direct product of a finite group and a free abelian group F and the rank of F is determined. A formula for the orthogonal idempotents of QG is found.

A special class of functional equations

2018 ◽  
Vol 68 (2) ◽  
pp. 397-404 ◽  
Author(s):  
Ahmed Charifi ◽  
Radosław Łukasik ◽  
Driss Zeglami
Keyword(s):  
Functional Equation ◽  
Finite Group ◽  
Abelian Group ◽  
Special Class ◽  
Functional Equations ◽  
Additive Functions ◽  
Group Of Automorphisms ◽  
Quadratic Equations ◽  
A Finite Group ◽  
Abelian Monoid

Abstract We obtain in terms of additive and multi-additive functions the solutions f, h: S → H of the functional equation $$\begin{array}{} \displaystyle \sum\limits_{\lambda \in \Phi }f(x+\lambda y+a_{\lambda })=Nf(x)+h(y),\quad x,y\in S, \end{array} $$ where (S, +) is an abelian monoid, Φ is a finite group of automorphisms of S, N = | Φ | designates the number of its elements, {aλ, λ ∈ Φ} are arbitrary elements of S and (H, +) is an abelian group. In addition, some applications are given. This equation provides a joint generalization of many functional equations such as Cauchy’s, Jensen’s, Łukasik’s, quadratic or Φ-quadratic equations.

scholarly journals On g-Noncommuting Graph of a Finite Group Relative to Its Subgroups

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3147
Author(s):  
Monalisha Sharma ◽  
Rajat Kanti Nath ◽  
Yilun Shang
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Dihedral Groups ◽  
Vertex Set ◽  
A Finite Group ◽  
Noncommuting Graph

Let H be a subgroup of a finite non-abelian group G and g∈G. Let Z(H,G)={x∈H:xy=yx,∀y∈G}. We introduce the graph ΔH,Gg whose vertex set is G\Z(H,G) and two distinct vertices x and y are adjacent if x∈H or y∈H and [x,y]≠g,g−1, where [x,y]=x−1y−1xy. In this paper, we determine whether ΔH,Gg is a tree among other results. We also discuss about its diameter and connectivity with special attention to the dihedral groups.

The metric dimension & distance spectrum of non-commuting graph of dihedral group

2021 ◽  
pp. 2150082
Author(s):  
Subarsha Banerjee
Keyword(s):  
Finite Group ◽  
Dihedral Group ◽  
Metric Dimension ◽  
Distance Spectrum ◽  
Commuting Graph ◽  
Vertex Set ◽  
A Finite Group

The non-commuting graph [Formula: see text] of a finite group [Formula: see text] has vertex set as [Formula: see text] and any two vertices [Formula: see text] are adjacent if [Formula: see text]. In this paper, we have determined the metric dimension and resolving polynomial of [Formula: see text], where [Formula: see text] is the dihedral group of order [Formula: see text]. The distance spectrum of [Formula: see text] has also been determined for all [Formula: see text].

RELATIVE NON-COMMUTING GRAPH OF A FINITE GROUP

2012 ◽  
Vol 12 (02) ◽  
pp. 1250157 ◽  
Author(s):  
B. TOLUE ◽  
A. ERFANIAN
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
The Other ◽  
Commuting Graph ◽  
Other Hand ◽  
A Finite Group

The essence of the non-commuting graph remind us to find a connection between this graph and the commutativity degree as denoted by d(G). On the other hand, d(H, G) the relative commutativity degree, was the key to generalize the non-commuting graph ΓG to the relative non-commuting graph (denoted by ΓH, G) for a non-abelian group G and a subgroup H of G. In this paper, we give some results about ΓH, G which are mostly new. Furthermore, we prove that if (H1, G1) and (H2, G2) are relative isoclinic then ΓH1, G1 ≅ Γ H2, G2 under special conditions.

Counting symmetric colorings of G × ℤ2

2019 ◽  
Vol 18 (10) ◽  
pp. 1950199
Author(s):  
Jabulani Phakathi ◽  
Shivani Singh ◽  
Yevhen Zelenyuk ◽  
Yuliya Zelenyuk
Keyword(s):  
Finite Group ◽  
Dihedral Group ◽  
A Finite Group

Let [Formula: see text] be a finite group and let [Formula: see text]. An [Formula: see text]-coloring of [Formula: see text] is any mapping [Formula: see text]. A coloring [Formula: see text] is symmetric if there is [Formula: see text] such that [Formula: see text] for every [Formula: see text]. We show that if [Formula: see text] is Abelian and [Formula: see text] is the polynomial representing the number of symmetric [Formula: see text]-colorings of [Formula: see text], then the number of symmetric [Formula: see text]-colorings of [Formula: see text] is [Formula: see text]. We also extend this result to the dihedral group [Formula: see text].

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