Recognition by noncommuting graph of finite simple groups L₄(q)

Abstract

Let G be a nonabelian group. We define the noncommuting graph ∇(G) of G as follows: its vertex set is G\Z(G), the set of non-central elements of G, and two different vertices x and y are joined by an edge if and only if x and y do not commute as elements of G, i.e., [x,y]\ne \mathrm{1}. We prove that if L ∈ {L₄(7), L₄(11), L₄(13), L₄(16), L₄(17)} and G is a finite group such that \nabla (G)\cong \nabla (L), then G\cong L.

Key words： noncommuting graph; spectrum; prime graph; projective special linear group L₄(q); recognition by noncommuting graph

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