Commuting Involution Graphs for 4-Dimensional Projective Symplectic Groups

Abstract

For a group G and X a subset of G the commuting graph of G on X, denoted by CðG; XÞ, is the graph whose vertex set is X with x; y 2 X joined by an edge if x 6 ¼ y and x and y commute. If the elements in X are involutions, then CðG; XÞ is called a commuting involution graph. This paper studies CðG; XÞ when G is a 4-dimensional projective symplectic group over a finite field and X a G-conjugacy class of involutions, determining the diameters and structure of the discs of these graphs.

A4-Graph of Finite Simple Groups

“…Currently, this style of studying the algebraic properties of groups is the most common. There is a remarkable number of researches in this area, see for example [1,2,3]. Assume that G is a finite groups and X is a conjugacy class of order 3 in G. In this work, we present the A 4 -graph of G as a simple graph denoted by A 4 (G,X).…”

“…The next examples are to illustrate the structure of A 4 -graph for certain finite groups. Examples 1.2 (1) Let GS 5 be a symmetric group of degree 5 and t= (3,4,5), then we have : X=t G =[ (3,4,5), (3,5,4), (2,3,4), (2,3,5), (2,4,3), (2,4,5), (2,5,3), (2,5,4), (1,2,3), (1,2,4), (1,2,5), (1,3,2), (1,3,4), (1,3,5), (1,4,2), (1,4,3), (1,…”

“…The graph A 4 (G,X) is connected with Diam A 4 (G,X) =3. The disc structures of the graph A 4 (G,X) are: (2,3,5), (2,4,3), (2,5,4), (1,3,5), (1,4,3), (1,5,4)},  2 (t)={ (2,3,4), (2,4,5), (2,5,3), (1,2,3), (1,2,4), (1,2,5), (1,3,2), (1,3,4), (1,4,2), (1,4,5), (1,5,2), (1,5,3) }, 3 (t)= {(3,5,4)}. This can be achieved computationally by using the gap package YAGS [7] as we describe in the next procedure which proceeds as follows: Procedure 1 1.…”

Commuting Involution Graphs for Certain Exceptional Groups of Lie Type

“…Suppose that G is a finite group and X is a subset of G. The commuting graph, CðG; XÞ, has X as its vertex set and two vertices x; y 2 X are joined by an edge if x 6 ¼ y and x and y commute. The extensive bibliography in [9] points towards the many varied commuting graphs which have been studied. But here we shall be considering commuting involution graphs-these are commuting graphs CðG; XÞ where X is a G-conjugacy class of involutions.…”

“…Recently the commuting involution graphs of the sporadic simple groups have received much attention, see [5,12,14,15,17]. For those simple groups of Lie type consult [1,4,[8][9][10], while an analysis of the commuting involution graphs of finite Coxeter groups may be found in [2,3].…”