scholarly journals Identity Graph of Finite Cyclic Groups

2021 ◽  
Vol 03 (01) ◽  
pp. 101-110
Author(s):  
Maria Vianney Any Herawati ◽  
◽  
Priscila Septinina Henryanti ◽  
Ricky Aditya ◽  
◽  
...  
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Identity Element ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Even Order ◽  
A Finite Group

This paper discusses how to express a finite group as a graph, specifically about the identity graph of a cyclic group. The term chosen for the graph is an identity graph, because it is the identity element of the group that holds the key in forming the identity graph. Through the identity graph, it can be seen which elements are inverse of themselves and other properties of the group. We will look for the characteristics of identity graph of the finite cyclic group, for both cases of odd and even order.

Some Remarks on Groups Admitting a Fixed-Point-Free Automorphism

1968 ◽  
Vol 20 ◽  
pp. 1300-1307 ◽  
Author(s):  
Fletcher Gross
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Simple Group ◽  
Direct Product ◽  
Finite Simple Group ◽  
Identity Element ◽  
Simple Groups ◽  
Cyclic Groups ◽  
A Finite Group ◽  
Open Question

A finite group G is said to be a fixed-point-free-group (an FPF-group) if there exists an automorphism a which fixes only the identity element of G. The principal open question in connection with these groups is whether non-solvable FPF-groups exist. One of the results of the present paper is that if a Sylow p-group of the FPF-group G is the direct product of any number of mutually non-isomorphic cyclic groups, then G has a normal p-complement. As a consequence of this, the conjecture that all FPF-groups are solvable would be true if it were true that every finite simple group has a non-trivial SylowT subgroup of the kind just described. Here it should be noted that all the known simple groups satisfy this property.

scholarly journals Homotopy Formulas for Cyclic Groups Acting on Rings

2008 ◽  
Vol 51 (1) ◽  
pp. 81-85
Author(s):  
Christian Kassel
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Cohomology Groups ◽  
Joint Work ◽  
Cyclic Groups ◽  
A Finite Group

AbstractThe positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff.

scholarly journals Finite groups whose intersection power graphs are toroidal and projective-planar

2021 ◽  
Vol 19 (1) ◽  
pp. 850-862
Author(s):  
Huani Li ◽  
Xuanlong Ma ◽  
Ruiqin Fu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Identity Element ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

Abstract The intersection power graph of a finite group G G is the graph whose vertex set is G G , and two distinct vertices x x and y y are adjacent if either one of x x and y y is the identity element of G G , or ⟨ x ⟩ ∩ ⟨ y ⟩ \langle x\rangle \cap \langle y\rangle is non-trivial. In this paper, we completely classify all finite groups whose intersection power graphs are toroidal and projective-planar.

scholarly journals On L h , k -Labeling Index of Inverse Graphs Associated with Finite Cyclic Groups

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
K. Mageshwaran ◽  
G. Kalaimurugan ◽  
Bussakorn Hammachukiattikul ◽  
Vediyappan Govindan ◽  
Ismail Naci Cangul
Keyword(s):  
Cyclic Group ◽  
Upper Bound ◽  
Labeling Index ◽  
Positive Difference ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
The Difference ◽  
Radio Labeling

An L h , k -labeling of a graph G = V , E is a function f : V ⟶ 0 , ∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k . The difference between the highest and lowest assigned values is the index of an L h , k -labeling. The minimum number for which the graph admits an L h , k -labeling is called the required possible index of L h , k -labeling of G , and it is denoted by λ k h G . In this paper, we obtain an upper bound for the index of the L h , k -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between L h , k -labeling with radio labeling of an inverse graph associated with a finite cyclic group.

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.

scholarly journals Automorphisms of Regular Wreath Product -Groups

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Representation Theory ◽  
Wreath Product ◽  
Cyclic Group ◽  
Product Group ◽  
Finite Cyclic Group ◽  
Arbitrary Finite Group

We present a useful new characterization of the automorphisms of the regular wreath product group of a finite cyclic -group by a finite cyclic -group, for any prime , and we discuss an application. We also present a short new proof, based on representation theory, for determining the order of the automorphism group Aut(), where is the regular wreath product of a finite cyclic -group by an arbitrary finite -group.

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.

Uniqueness of Free Actions on S3 Respecting a Knot

1987 ◽  
Vol 39 (4) ◽  
pp. 969-982 ◽  
Author(s):  
Michel Boileau ◽  
Erica Flapan
Keyword(s):  
Fixed Point ◽  
Cyclic Group ◽  
Group Action ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Cyclic Group Action ◽  
Periodic Diffeomorphisms ◽  
Free Actions

In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.

Skew-morphisms of cyclic 2-groups

2019 ◽  
Vol 22 (4) ◽  
pp. 617-635
Author(s):  
Shaofei Du ◽  
Kan Hu
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Identity Element ◽  
A Finite Group

AbstractA skew-morphism of a finite group A is a permutation φ on A fixing the identity element, and for which there exists an integer function π on A such that, for all {x,y\in A}, {\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)}. In [I. Kovács and R. Nedela, Skew-morphisms of cyclic p-groups, J. Group Theory 20 2017, 6, 1135–1154], Kovács and Nedela determined skew-morphisms of the cyclic p-groups for any odd prime p. In this paper, we shall determine that of cyclic 2-groups.

scholarly journals On the minimum degree of the power graph of a finite cyclic group

2020 ◽  
pp. 2150044
Author(s):  
Ramesh Prasad Panda ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Finite Groups ◽  
Minimum Degree ◽  
Simple Graph ◽  
The Other ◽  
Prime Divisors ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

The power graph [Formula: see text] of a finite group [Formula: see text] is the undirected simple graph whose vertex set is [Formula: see text], in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. The minimum degree [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For [Formula: see text], under certain conditions involving the prime divisors of [Formula: see text], we identify at most [Formula: see text] vertices such that [Formula: see text] is equal to the degree of at least one of these vertices. If [Formula: see text], or that [Formula: see text] is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of [Formula: see text].

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