scholarly journals On the dominating number, independent number and the regularity of the relative co-prime graph of a group

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Norarida Abd Rhani ◽  
Nor Muhainiah Mohd Ali ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Finite Group ◽  
Dominating Set ◽  
Prime Graph ◽  
Greatest Common Divisor ◽  
Dominating Number ◽  
Independent Number ◽  
A Finite Group

Let H be a subgroup of a finite group G. 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 of x and y is equal to one. This concept has been extended to the relative co-prime graph of a group with respect to a subgroup H, which is defined as a graph whose vertices are elements of G and two distinct vertices x and y are joined by an edge if and only if their orders are co-prime and any of x or y is in H.  Some properties of graph such as the dominating number, degree of a dominating set of order one and independent number are obtained. Lastly, the regularity of the relative co-prime graph of a group is found.

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.

n-RECOGNITION BY PRIME GRAPH OF THE SIMPLE GROUP PSL(2,q)

2008 ◽  
Vol 07 (06) ◽  
pp. 735-748 ◽  
Author(s):  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Number ◽  
Finite Groups ◽  
Prime Graph ◽  
Split Extension ◽  
A Finite Group

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge if there is an element in G of order pq. It is proved that if p > 11 and p ≢ 1 (mod 12), then PSL(2,p) is uniquely determined by its prime graph. Also it is proved that if p > 7 is a prime number and Γ(G) = Γ(PSL(2,p2)), then G ≅ PSL(2,p2) or G ≅ PSL(2,p2).2, the non-split extension of PSL(2,p2) by ℤ2. In this paper as the main result we determine finite groups G such that Γ(G) = Γ(PSL(2,q)), where q = pk. As a consequence of our results we prove that if q = pk, k > 1 is odd and p is an odd prime number, then PSL(2,q) is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.

CHARACTERIZATION BY PRIME GRAPH OF PGL(2, pk) WHERE p AND k > 1 ARE ODD

2010 ◽  
Vol 20 (07) ◽  
pp. 847-873 ◽  
Author(s):  
Z. AKHLAGHI ◽  
B. KHOSRAVI ◽  
M. KHATAMI
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Prime Number ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Prime Graphs ◽  
A Finite Group ◽  
Nonabelian Composition Factor

Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 71–85], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs.

The prime graph on class sizes of a finite group has a bipartite complement

2020 ◽  
Vol 542 ◽  
pp. 35-42
Author(s):  
Silvio Dolfi ◽  
Emanuele Pacifici ◽  
Lucia Sanus ◽  
Víctor Sotomayor
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
A Finite Group

scholarly journals On P. Hall's Generalisation of a Theorem of Frobenius

1962 ◽  
Vol 5 (3) ◽  
pp. 97-100 ◽  
Author(s):  
S. K. Sehgal
Keyword(s):  
Finite Group ◽  
Greatest Common Divisor ◽  
Number Of Solutions ◽  
Fixed Element ◽  
A Finite Group

1. It is a well known theorem due to Frobenius that the number of solutions of the equationXn = 1in a finite group G, is a multiple of the greatest common divisor (n, g) of n and the order g of G. Frobenius himself proved later that the number of solutions of the equationXn = awhere a is a fixed element of G, is a multiple of (n, ga), ga being the order of the centralizer Z(a) of ain G.

scholarly journals On alternating and symmetric groups which are quasi OD-characterizable

2017 ◽  
Vol 16 (04) ◽  
pp. 1750065 ◽  
Author(s):  
Ali Reza Moghaddamfar
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Prime Graph ◽  
Symmetric Groups ◽  
Degree Pattern ◽  
Alternating And Symmetric Groups ◽  
A Finite Group

Let [Formula: see text] be the prime graph associated with a finite group [Formula: see text] and [Formula: see text] be the degree pattern of [Formula: see text]. A finite group [Formula: see text] is said to be [Formula: see text]-fold [Formula: see text]-characterizable if there exist exactly [Formula: see text] nonisomorphic groups [Formula: see text] such that [Formula: see text] and [Formula: see text]. The purpose of this paper is two-fold. First, it shows that the symmetric group [Formula: see text] is [Formula: see text]-fold [Formula: see text]-charaterizable. Second, it shows that there exist many infinite families of alternating and symmetric groups, [Formula: see text] and [Formula: see text], which are [Formula: see text]-fold [Formula: see text]-characterizable with [Formula: see text].

A NEW CHARACTERIZATION OF ALMOST SPORADIC GROUPS

2002 ◽  
Vol 01 (03) ◽  
pp. 267-279 ◽  
Author(s):  
AMIR KHOSRAVI ◽  
BEHROOZ KHOSRAVI
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Sporadic Groups ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Order Components ◽  
Positive Integers ◽  
Coprime Positive Integers

Let G be a finite group. Based on the prime graph of G, the order of G can be divided into a product of coprime positive integers. These integers are called order components of G and the set of order components is denoted by OC(G). Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper we prove that almost sporadic simple groups, except Aut (J2) and Aut (McL), and the automorphism group of PSL(2, 2n) where n=2sare also uniquely determined by their order components. Also we discuss about the characterizability of Aut (PSL(2, q)). As corollaries of these results, we generalize a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.

scholarly journals The Lengths of Certain Real Conjugacy Classes and the Related Prime Graph

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2060
Author(s):  
Siqiang Yang ◽  
Xianhua Li
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Prime Graph ◽  
Conjugacy Classes ◽  
New Type ◽  
A Finite Group

Let G be a finite group. In this paper, we study how certain arithmetical conditions on the conjugacy class lengths of real elements of G influence the structure of G. In particular, a new type of prime graph is introduced and studied. We obtain a series of theorems which generalize some existed results.

A Characterization of Finite Simple Groups by the Degrees of Vertices of Their Prime Graphs

2005 ◽  
Vol 12 (03) ◽  
pp. 431-442 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
A. R. Zokayi ◽  
M. R. Darafsheh
Keyword(s):  
Finite Group ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Natural Numbers ◽  
Prime Graphs ◽  
Sporadic Simple Groups ◽  
A Finite Group ◽  
Groups Of Lie Type

If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if there is an element in G of order pq. Assume [Formula: see text] with primes p1<p2<⋯<pkand natural numbers αi. For p∈π(G), let the degree of p be deg (p)=|{q∈π(G)|q~p}|, and D(G):=( deg (p1), deg (p2),…, deg (pk)). In this paper, we prove that if G is a finite group such that D(G)=D(M) and |G|=|M|, where M is one of the following simple groups: (1) sporadic simple groups, (2) alternating groups Apwith p and p-2 primes, (3) some simple groups of Lie type, then G≅M. Moreover, we show that if G is a finite group with OC (G)={29.39.5.7, 13}, then G≅S6(3) or O7(3), and finally, we show that if G is a finite group such that |G|=29.39.5.7.13 and D(G)=(3,2,2,1,0), then G≅S6(3) or O7(3).

scholarly journals Quasirecognition by Prime Graph of the Groups 2D2n(q) Where q < 105

2017 ◽  
Author(s):  
Hossein Moradi ◽  
Mohammad Reza Darafsheh ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Prime Power ◽  
Prime Graph ◽  
Composition Factor ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Prime Divisors ◽  
A Finite Group

Let G be a finite group. The prime graph &Gamma;(G) of G is defined as follows: The set of vertices of&nbsp;&Gamma;(G) is the set of prime divisors of |G| and two distinct vertices p and p' are connected in &Gamma;(G), whenever G has an element of order pp'. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with &Gamma;(G)=&Gamma;(P), G has a composition factor isomorphic to P. In&nbsp;[4] proved finite simple groups 2Dn(q), where&nbsp;n&nbsp;&ne; 4k are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups 2D2k(q), where&nbsp;k &ge; 9 and q is a prime power less than 105.

Sign in / Sign up

Export Citation Format

Share Document