Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Commuting Involution Graphs for 3-Dimensional Unitary Groups. | Commuting Involution Graphs for 3-Dimensional Unitary Groups Alistaire Everett School of Mathematics The University of Manchester Oxford Road Manchester M13 9PL UK Submitted Feb 7 2011 Accepted Apr 28 2011 Published May 8 2011 Mathematics Subject Classification 05C12 20E99 Abstract For a group G and X a subset of G the commuting graph of G on X denoted by C G X is the graph whose vertex set is X with x y E X joined by an edge if x y and x and y commute. If the elements in X are involutions then C G X is called a commuting involution graph. This paper studies 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. 1 Introduction For a group G and a subset X of G we define the commuting graph denoted C G X to be the graph whose vertex set is X with two distinct vertices x y E X joined by an edge if and only if xy yx. Commuting graphs first came to prominence in the groundbreaking paper of Brauer and Fowler 6 famous for containing a proof that only finitely many finite simple groups can contain a given involution centralizer. The commuting graphs employed in this paper had X G 1 - such graphs have played a vital role in recent results relating to the Margulis-Platanov conjecture see 11 . When X is a conjugacy class of involutions we call C G X a commuting involution graph. This special case demonstrated its importance in the mostly unpublished work of Fischer 9 which led to the construction of three new sporadic simple groups. Aschbacher 1 also showed a necessary condition on a commuting involution graph for the presence of a strongly embedded subgroup in G. The detailed study of commuting involution graphs came to the fore in 2003 with the work of Bates Bundy Hart nee Perkins and Rowley which explored commuting involution graphs for G a symmetric group or more generally a finite Coxeter group a special linear group .
