THE SQUARED COMMUTATIVITY DEGREE OF DIHEDRAL GROUPS

2016 ◽  
Vol 78 (3-2) ◽  
Author(s):  
Muhanizah Abdul Hamid ◽  
Nor Muhainiah Mohd Ali ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian ◽  
Fadila Normahia Abd Manaf
Keyword(s):  
Finite Group ◽  
Dihedral Groups ◽  
A Finite Group ◽  
Random Pair

The commutativity degree of a finite group is the probability that a random pair of elements in the group commute. Furthermore, the n-th power commutativity degree of a group is a generalization of the commutativity degree of a group which is defined as the probability that the n-th power of a random pair of elements in the group commute. In this paper, the n-th power commutativity degree for some dihedral groups is computed for the case n equal to 2, called the squared commutativity degree.

scholarly journals Co-prime Probability for Nonabelian Metabelian Groups of Order Less than 24 and Their Related Graphs

MATEMATIKA ◽  
2019 ◽  
Vol 35 (3) ◽  
Author(s):  
Nurfarah Zulkifli ◽  
Nor Muhainiah Mohd Ali
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Greatest Common Divisor ◽  
Metabelian Groups ◽  
A Finite Group ◽  
Random Pair

Let G be a finite group. The probability of a random pair of elements in G are said to be co-prime when the greatest common divisor of order x and y, where x and y in G, is equal to one. Meanwhile the co-prime graph of a group is defined as a graph whose vertices are elements of G and two distinct vertices are adjacent if and only if the greatest common divisor of order x and y is equal to one. In this paper, the co-prime probability and its graph such as the type and the properties of the graph are determined.

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.

scholarly journals On product-one sequences over dihedral groups

2020 ◽  
pp. 2250064
Author(s):  
Alfred Geroldinger ◽  
David J. Grynkiewicz ◽  
Jun Seok Oh ◽  
Qinghai Zhong
Keyword(s):  
Finite Group ◽  
Identity Element ◽  
Finite Sequence ◽  
Finitely Generated ◽  
Dihedral Groups ◽  
Group A ◽  
Arithmetic Combinatorics ◽  
A Finite Group ◽  
Extremal Properties

Let [Formula: see text] be a finite group. A sequence over [Formula: see text] means a finite sequence of terms from [Formula: see text], where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over [Formula: see text] (with the concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This paper provides a detailed investigation, with methods from arithmetic combinatorics, of the arithmetic of the monoid of product-one sequences over dihedral groups.

scholarly journals COMMUTATOR EQUATIONS IN FINITE GROUPS

2021 ◽  
pp. 1-16
Author(s):  
KANTO IRIMOTO ◽  
ENRIQUE TORRES-GIESE
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Probability Distributions ◽  
Group Element ◽  
Solution Set ◽  
System Of Equations ◽  
Systems Of Equations ◽  
Dihedral Groups ◽  
Natural Extensions ◽  
A Finite Group

Abstract The problem of finding the number of ordered commuting tuples of elements in a finite group is equivalent to finding the size of the solution set of the system of equations determined by the commutator relations that impose commutativity among any pair of elements from an ordered tuple. We consider this type of systems for the case of ordered triples and express the size of the solution set in terms of the irreducible characters of the group. The obtained formulas are natural extensions of Frobenius’ character formula that calculates the number of ways a group element is a commutator of an ordered pair of elements in a finite group. We discuss how our formulas can be used to study the probability distributions afforded by these systems of equations, and we show explicit calculations for dihedral groups.

scholarly journals Total characters and Chebyshev polynomials

2003 ◽  
Vol 2003 (38) ◽  
pp. 2447-2453 ◽  
Author(s):  
Eirini Poimenidou ◽  
Homer Wolfe
Keyword(s):  
Finite Group ◽  
Chebyshev Polynomials ◽  
Irreducible Character ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Irreducible Characters ◽  
Integer Coefficients ◽  
Complete Answer ◽  
A Finite Group

The total characterτof a finite groupGis defined as the sum of all the irreducible characters ofG. K. W. Johnson asks when it is possible to expressτas a polynomial with integer coefficients in a single irreducible character. In this paper, we give a complete answer to Johnson's question for all finite dihedral groups. In particular, we show that, when such a polynomial exists, it is unique and it is the sum of certain Chebyshev polynomials of the first kind in any faithful irreducible character of the dihedral groupG.

scholarly journals Regulator constants of integral representations of finite groups

2018 ◽  
Vol 168 (1) ◽  
pp. 75-117 ◽  
Author(s):  
ALEX TORZEWSKI
Keyword(s):  
Finite Group ◽  
Elliptic Curves ◽  
Finite Groups ◽  
Isomorphism Class ◽  
Integral Representations ◽  
Dihedral Groups ◽  
Representation Ring ◽  
Representations Of Finite Groups ◽  
Cohomological Invariant ◽  
A Finite Group

AbstractLet G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

scholarly journals Lie theory of finite simple groups and the Roth property

2017 ◽  
Vol 163 (2) ◽  
pp. 301-340 ◽  
Author(s):  
J. LÓPEZ PEÑA ◽  
S. MAJID ◽  
K. RIETSCH
Keyword(s):  
Finite Group ◽  
Dimensional Subspace ◽  
Conjugacy Classes ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Killing Form ◽  
Dihedral Groups ◽  
Invariant Vectors ◽  
A Finite Group ◽  
Conjugation Representation

AbstractIn noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset $\mathcal{C}$ stable under inversion. We study the associated Killing form K. For the universal calculus associated to $\mathcal{C}$ = G \ {e} we show that the magnitude $\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$ of the Killing form is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ ≠ 1/(N − 1) where N is the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N − 1)-dimensional subspace of invariant vectors is invertible iff the finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in any Sn has invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groups D2n with n odd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

On transiso graph

2015 ◽  
Vol 08 (04) ◽  
pp. 1550070 ◽  
Author(s):  
Vipul Kakkar ◽  
Laxmi Kant Mishra
Keyword(s):  
Finite Group ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Dihedral Groups ◽  
A Finite Group

In this paper, we define a new graph [Formula: see text] on a finite group [Formula: see text], where [Formula: see text] is a divisor of [Formula: see text]. The vertices of [Formula: see text] are the subgroups of [Formula: see text] of order [Formula: see text] and two subgroups [Formula: see text] and [Formula: see text] of [Formula: see text] are said to be adjacent if there exists [Formula: see text] [Formula: see text] such that [Formula: see text], where [Formula: see text] [Formula: see text] denote the set of all NRTs of [Formula: see text] in [Formula: see text]. We shall discuss the completeness of [Formula: see text] for various groups like finite abelian groups, dihedral groups and some finite [Formula: see text]-groups.

scholarly journals Which Haar graphs are Cayley graphs?

10.37236/5240 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
István Estélyi ◽  
Tomaž Pisanski
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Cayley Graph ◽  
Regular Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Equivalent Condition ◽  
Dihedral Groups ◽  
A Finite Group

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containing $G$ is derived in terms of $G, S$ and $\mathrm{Aut } G$. It is also shown that the dihedral groups, which are solutions to the above problem, are $\mathbb{Z}_2^2,D_3,D_4$ and $D_{5}$. 

Laplacian spectra of Coprime Graph of finite cyclic and Dihedral groups

2020 ◽  
pp. 2150020
Author(s):  
Subarsha Banerjee
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Characteristic Polynomial ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Vertex Connectivity ◽  
Laplacian Eigenvalues ◽  
Laplacian Spectra ◽  
A Finite Group ◽  
Second Largest Eigenvalue

Let [Formula: see text] denote the Coprime Graph of a finite group [Formula: see text]. In this paper we study the Laplacian eigenvalues of the Coprime Graph of the finite cyclic group [Formula: see text] and the Dihedral group [Formula: see text] where [Formula: see text]. We find the characteristic polynomial of [Formula: see text] for any [Formula: see text] and determine the eigenvalues of [Formula: see text] for [Formula: see text] where [Formula: see text] are primes and [Formula: see text] is a positive integer. We characterize the values of [Formula: see text] for which algebraic and vertex connectivity of [Formula: see text] are equal. We also discuss about the largest and the second largest eigenvalue of [Formula: see text]. Finally, the spectra of [Formula: see text] has been determined for [Formula: see text] where [Formula: see text] are primes and [Formula: see text] is a positive integer.

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