# 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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#### References

A. Abdollahi, S. Akbari and H. R. Maimani (2006). Non-commting graph of a group. J. Algebra. 298, 468-492 E. Artin (1955). The orders of the linear groups. Comm. Pure Appl. Math.. 8, 355-365 E. Artin (1955). The orders of the classical simple groups. Comm. Pure Appl. Math.. 8, 455-472 M. R. Darafsheh (2009). Groups with the same non-commutng graph. Discrete Appl. Math.. 157 (4), 833-837 M. R. Darafsheh and P. Yousefzadeh A characterization of the group $BA_{22}$ by non-commuting graph. to appear in Bull. Korean Math. Soc.. R. Solomon and A. Woldar (2012). All simple groups are charcterized by their non-commuting graphs. preprint. S. H. Alavi and A. Daneshkhah (2005). A new characterization of alternating and symmetric groups. J. Appl. Math. Comput.. 17, 245-258