scholarly journals There is no upper bound for the diameter of the commuting graph of a finite group

2013 ◽  
Vol 120 (7) ◽  
pp. 1600-1603 ◽  
Author(s):  
Michael Giudici ◽  
Chris Parker
Keyword(s):  
Finite Group ◽  
Upper Bound ◽  
Commuting Graph ◽  
A Finite Group

Conditions on the Edges and Vertices of Non-commuting Graph

2015 ◽  
Vol 74 (1) ◽  
Author(s):  
M. Jahandideh ◽  
M. R. Darafsheh ◽  
N. H. Sarmin ◽  
S. M. S. Omer
Keyword(s):  
Finite Group ◽  
Conjugacy Classes ◽  
Commuting Graph ◽  
Vertex Set ◽  
A Finite Group

Abstract - Let G􀡳 be a non- abelian finite group. The non-commuting graph ,􀪡is defined as a graph with a vertex set􀡳 − G-Z(G)􀢆in which two vertices x􀢞 and y􀢟 are joined if and only if xy􀢞􀢟 ≠ yx􀢟􀢞.  In this paper, we invest some results on the number of edges set , the degree of avertex of non-commuting graph and the number of conjugacy classes of a finite group. In order that if 􀪡􀡳non-commuting graph of H ≅ non - commuting graph of G􀪡􀡴,H 􀡴 is afinite group, then |G􀡳| = |H􀡴| .

On the order of the automorphism group of a finite group. II

1956 ◽  
Vol 235 (1201) ◽  
pp. 235-246 ◽  
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Upper Bound ◽  
Group Ii ◽  
A Finite Group

Corresponding to a fixed prime p there exists a function g(h) such that the order of the automorphism group of a finite group G is divisible by p h , provided that the order of G is divisible by p g(h) . An explicit upper bound for g(h) is obtained.

scholarly journals On Diameter of Subgraphs of Commuting Graph in Symplectic Group for Elements of Order Three

2021 ◽  
Vol 50 (2) ◽  
pp. 549-557
Author(s):  
Suzila Mohd Kasim ◽  
Athirah Nawawi
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Symplectic Group ◽  
Undirected Graph ◽  
The Other ◽  
Symplectic Groups ◽  
Main Work ◽  
Commuting Graph ◽  
Mathematical Formulas ◽  
A Finite Group

Suppose G be a finite group and X be a subset of G. The commuting graph, denoted by C(G,X), is a simple undirected graph, where X ⊂G being the set of vertex and two distinct vertices x,y∈X are joined by an edge if and only if xy = yx. The aim of this paper was to describe the structure of disconnected commuting graph by considering a symplectic group and a conjugacy class of elements of order three. The main work was to discover the disc structure and the diameter of the subgraph as well as the suborbits of symplectic groups S4(2)', S4(3) and S6(2). Additionally, two mathematical formulas are derived and proved, one gives the number of subgraphs based on the size of each subgraph and the size of the conjugacy class, whilst the other one gives the size of disc relying on the number and size of suborbits in each disc.

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.

ON AN INVERSE PROBLEM TO FROBENIUS' THEOREM II

2012 ◽  
Vol 11 (05) ◽  
pp. 1250092 ◽  
Author(s):  
WEI MENG ◽  
JIANGTAO SHI ◽  
KELIN CHEN
Keyword(s):  
Inverse Problem ◽  
Finite Group ◽  
Positive Integer ◽  
Upper Bound ◽  
Complete Classification ◽  
Frobenius Theorem ◽  
A Finite Group

Let G be a finite group and e a positive integer dividing |G|, the order of G. Denoting Le(G) = {x ∈ G|xe = 1}. Frobenius proved that |Le(G)| = ke for some positive integer k ≥ 1. Let k(G) be the upper bound of the set {k||Le(G)| = ke, ∀ e ||G|}. In this paper, a complete classification of the finite group G with k(G) = 3 is obtained.

scholarly journals B.H. Neumann's Question on Ensuring Commutativity of Finite Groups

2006 ◽  
Vol 74 (1) ◽  
pp. 121-132 ◽  
Author(s):  
A. Abdollahi ◽  
A. Azad ◽  
A. Mohammadi Hassanabadi ◽  
M. Zarrin
Keyword(s):  
Finite Group ◽  
Exponential Function ◽  
Finite Groups ◽  
Upper Bound ◽  
Partial Answer ◽  
A Finite Group

This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: “Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?” We find an exponential function f(m,n) such that every such group G is Abelian whenever |G| > f(m,n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble.

scholarly journals Automorphisms of Finite Groups and their Fixed-Point Groups

1969 ◽  
Vol 9 (3-4) ◽  
pp. 467-477 ◽  
Author(s):  
J. N. Ward
Keyword(s):  
Fixed Point ◽  
Finite Group ◽  
Abelian Group ◽  
Finite Groups ◽  
Upper Bound ◽  
Prime Order ◽  
Soluble Group ◽  
Point Groups ◽  
Derived Series ◽  
A Finite Group

Let G denote a finite group with a fixed-point-free automorphism of prime order p. Then it is known (see [3] and [8]) that G is nilpotent of class bounded by an integer k(p). From this it follows that the length of the derived series of G is also bounded. Let l(p) denote the least upper bound of the length of the derived series of a group with a fixed-point-free automorphism of order p. The results to be proved here may now be stated: Theorem 1. Let G denote a soluble group of finite order and A an abelian group of automorphisms of G. Suppose that (a) ∣G∣ is relatively prime to ∣A∣; (b) GAis nilpotent and normal inGω, for all ω ∈ A#; (c) the Sylow 2-subgroup of G is abelian; and (d) if q is a prime number andqk+ 1 divides the exponent of A for some integer k then the Sylow q-subgroup of G is abelian.

On the Diameters of Commuting Graphs Arising from Random Skew-Symmetric Matrices

2014 ◽  
Vol 23 (3) ◽  
pp. 449-459 ◽  
Author(s):  
PETER HEGARTY ◽  
DMITRII ZHELEZOV
Keyword(s):  
Finite Group ◽  
Random Graph ◽  
Large Family ◽  
Parameter Family ◽  
Independent Interest ◽  
Symmetric Matrices ◽  
Commuting Graph ◽  
A Finite Group ◽  
Random Groups ◽  
Two Parameter

We present a two-parameter family $(G_{m,k})_{m, k \in \mathbb{N}_{\geq 2}}$, of finite, non-abelian random groups and propose that, for each fixed k, as m → ∞ the commuting graph of Gm,k is almost surely connected and of diameter k. We present heuristic arguments in favour of this conjecture, following the lines of classical arguments for the Erdős–Rényi random graph. As well as being of independent interest, our groups would, if our conjecture is true, provide a large family of counterexamples to the conjecture of Iranmanesh and Jafarzadeh that the commuting graph of a finite group, if connected, must have a bounded diameter. Simulations of our model yielded explicit examples of groups whose commuting graphs have all diameters from 2 up to 10.

scholarly journals THE AUTOMORPHISM GROUP OF COMMUTING GRAPH OF A FINITE GROUP

2014 ◽  
Vol 51 (4) ◽  
pp. 1145-1153 ◽  
Author(s):  
Mahsa Mirzargar ◽  
Peter P. Pach ◽  
A.R. Ashrafi
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Commuting Graph ◽  
A Finite Group
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