Characterization of the symmetric group by its non-commuting graph

Document Type: Research Paper

Authors

1 University of Tehran

2 K. N. Toosi University of Technology

Abstract

‎The non-commuting graph $\nabla(G)$ of a non-abelian group $G$ is defined as‎ ‎follows‎: ‎its vertex set is $G-Z(G)$ and two distinct vertices $x$ and $y$ are‎ ‎joined by an edge if and only if the commutator of $x$ and $y$ is not the‎ ‎identity‎. ‎In this paper we prove that if $G$ is a finite group with‎ ‎$\nabla(G) \cong \nabla(BS_n)$‎, ‎then $G \cong BS_n$‎, ‎where $BS_n$‎ ‎is the symmetric group of degree $n$‎, ‎where $n$ is a natural number‎.

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