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Unformatted text preview: Commuting graphs of odd prime order elements in simple groups * B. Baumeister, A. Stein December 30, 2009 Abstract We study the commuting graph on elements of odd prime order in finite simple groups. The results are used in a forthcoming paper describing the structure of finite Bruck loops and of Bol loops of exponent 2. 1 Introduction Let G be a group and X a normal subset of G , that is for all x ∈ X,g ∈ G we have x g ∈ X . The commuting graph on X is the undirected graph Γ X,G = Γ X with vertex set X such that two different vertices x and y are adjacent if and only if [ x,y ] = 1. The commuting graph of a group is an object which has been studied quite often to obtain strong results on the group G . We give a short overview of some major work on or related to commuting graphs. For more details see the references given below. Bender noted in his paper on strongly 2-embedded subgroups, [B], the equiv- alence between the existence of a strongly 2-embedded subgroup and the dis- connectedness of the commuting graph of involutions. At about the same time Fischer determined the groups generated by a class X of 3-transpositions by studying the commuting graph on X [Fi]. Later Stell- macher classified those groups which are generated by a special class of elements of order 3 again by examining the related commuting graph [St]. To prove the uniqueness of the sporadic simple group Ly, Aschbacher and Segev showed that its commuting graph on 3-central elements is simply con- nected [AS]. In addition, a major breakthrough towards the famous Margulis- Platonov conjecture has been made by Segev by using the commuting graph on the whole set G for G a non-trivial finite group [Se]. Finally Bates et al. [BBPR] determined the diameter of the connected com- muting graphs of a conjugacy class of involutions of G where G is a Coxeter group and Perkins [Pe] did the same for the affine groups ˜ A n , see also the related work [IJ2]. In [AAM] Abdollahi, Akbari and Maimanithe considered the dual of the commuting graph on G \ Z ( G ). They conjectured that if these graphs are isomorphic for two non-abelian finite groups then the groups have the same order. This conjecture has been checked for some simple groups in [IJ1]. * This research is part of the project “Transversals in Groups with an application to loops” GZ: BA 2200/2-2 funded by the DFG 1 In this paper we study the connected components of the commuting graph Γ O on the set O of odd prime order elements of a finite simple group G and of some of its subgraphs. We use our main results in [S] and [BS] to characterize the finite Bruck loops of 2-power exponent. We also consider our theorems to be of independent interest....
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This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell.
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