scholarly journals Commuting Involution Graphs for $3$-Dimensional Unitary Groups

10.37236/590 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Alistaire Everett
Keyword(s):  
Conjugacy Class ◽  
Unitary Group ◽  
Unitary Groups ◽  
Special Unitary Group ◽  
3 Dimensional ◽  
Commuting Graph ◽  
Vertex Set

For a group $G$ and $X$ a subset of $G$ the commuting graph of $G$ on $X$, denoted by $\cal{C}(G,X)$, is the graph whose vertex set is $X$ with $x,y\in X$ joined by an edge if $x\neq y$ and $x$ and $y$ commute. If the elements in $X$ are involutions, then $\cal{C}(G,X)$ is called a commuting involution graph. This paper studies $\cal{C}(G,X)$ when $G$ is a 3-dimensional projective special unitary group and $X$ a $G$-conjugacy class of involutions, determining the diameters and structure of the discs of these graphs.

scholarly journals Real classes of finite special unitary groups

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Amanda A. Schaeffer Fry ◽  
C. Ryan Vinroot
Keyword(s):  
Unitary Group ◽  
Unitary Groups ◽  
Special Unitary Group

AbstractWe classify all real and strongly real classes of the finite special unitary group

scholarly journals A Note on Commuting Graphs for Symmetric Groups

10.37236/95 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
C. Bates ◽  
D. Bundy ◽  
S. Hart ◽  
P. Rowley
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Vertex Set

The commuting graph ${\cal C}(G,X)$, where $G$ is a group and $X$ a subset of $G$, has $X$ as its vertex set with two distinct elements of $X$ joined by an edge when they commute in $G$. Here the diameter and disc structure of ${\cal C}(G,X)$ is investigated when $G$ is the symmetric group and $X$ a conjugacy class of $G$.

scholarly journals On Commuting Graphs for Elements of Order 3 in Symmetric Groups

10.37236/2362 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Athirah Nawawi ◽  
Peter Rowley
Keyword(s):  
Conjugacy Class ◽  
Symmetric Group ◽  
Symmetric Groups ◽  
Commuting Graph ◽  
Vertex Set

The commuting graph $\mathcal{C}(G,X)$, where $G$ is a group and $X$ is a subset of $G$, is the graph with vertex set $X$ and distinct vertices being joined by an edge whenever they commute. Here the diameter of $\mathcal{C}(G,X)$ is studied when $G$ is a symmetric group and $X$ a conjugacy class of elements of order $3$.

scholarly journals Explicit Calculations of Automorphic Forms for Definite Unitary Groups

2008 ◽  
Vol 11 ◽  
pp. 326-342 ◽  
Author(s):  
David Loeffler
Keyword(s):  
Unitary Group ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Level Structure ◽  
Unitary Groups ◽  
3 Dimensional ◽  
Local Constancy

I give an algorithm for computing the full space of automor-phic forms for definite unitary groups over ℚ, and apply this to calculate the automorphic forms of level G(hat{Z}) and various small weights for an example of a rank 3 unitary group. This leads to some examples of various types of endoscopic lifting from automorphic forms for U1 × U1 × U1 and U1 × U2, and to an example of a non-endoscopic form of weight (3, 3) corresponding to a family of 3-dimensional irreducible ℓ-adic Galois representations. I also compute the 2-adic slopes of some automorphic forms with level structure at 2, giving evidence for the local constancy of the slopes.

Non-commuting graphs of certain almost simple groups

2019 ◽  
Vol 12 (05) ◽  
pp. 1950081
Author(s):  
M. Jahandideh ◽  
R. Modabernia ◽  
S. Shokrolahi
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Simple Group ◽  
Simple Groups ◽  
Almost Simple Group ◽  
Almost Simple Groups ◽  
Commuting Graph ◽  
Vertex Set

Let [Formula: see text] be a non-abelian finite group and [Formula: see text] be the center of [Formula: see text]. The non-commuting graph, [Formula: see text], associated to [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. We conjecture that if [Formula: see text] is an almost simple group and [Formula: see text] is a non-abelian finite group such that [Formula: see text], then [Formula: see text]. Among other results, we prove that if [Formula: see text] is a certain almost simple group and [Formula: see text] is a non-abelian group with isomorphic non-commuting graphs, then [Formula: see text].

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􀡴| .

scholarly journals The Non-Commuting, Non-Generating Graph of a Nilpotent Group

10.37236/9802 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Peter Cameron ◽  
Saul Freedman ◽  
Colva Roney-Dougal
Keyword(s):  
Maximal Subgroup ◽  
Nilpotent Group ◽  
Commuting Graph ◽  
Generating Graph ◽  
Central Elements ◽  
Vertex Set ◽  
Infinite Case ◽  
The Difference ◽  
The Relationship

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail.

scholarly journals A SURVEY ON THE AUTOMORPHISM GROUPS OF THE COMMUTING GRAPHS AND POWER GRAPHS

2019 ◽  
pp. 729
Author(s):  
Mahsa Mirzargar
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
The Other ◽  
Power Graph ◽  
Commuting Graph ◽  
Vertex Set ◽  
Delta G ◽  
Nite Group

Let G be a nite group. The power graph P(G) of a group G is the graphwhose vertex set is the group elements and two elements are adjacent if one is a power of the other. The commuting graph \Delta(G) of a group G, is the graph whose vertices are the group elements, two of them joined if they commute. When the vertex set is G-Z(G), this graph is denoted by \Gamma(G). Since the results based on the automorphism group of these kinds of graphs are so sporadic, in this paper, we give a survey of all results on the automorphism group of power graphs and commuting graphs obtained in the literature.

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