Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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THE AUTOMORPHISM GROUP OF COMMUTING GRAPH OF A FINITE GROUP

  • Mirzargar, Mahsa (Department of Pure Mathematics Faculty of Mathematical Sciences University of Kashan) ;
  • Pach, Peter P. (Department of Algebra and Number Theory Eotvos Lorand University) ;
  • Ashrafi, A.R. (Department of Pure Mathematics Faculty of Mathematical Sciences University of Kashan)
  • Received : 2013.09.12
  • Published : 2014.07.31
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Abstract

Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and $x,y{\in}X$ ($x{\neq}y$) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by ${\Delta}(G)$. The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(${\Delta}(G)$) is abelian if and only if ${\mid}G{\mid}{\leq}2$; ${\mid}Aut({\Delta}(G)){\mid}$ is of prime power if and only if ${\mid}G{\mid}{\leq}2$, and ${\mid}Aut({\Delta}(G)){\mid}$ is square-free if and only if ${\mid}G{\mid}{\leq}3$. Some new graphs that are useful in studying the automorphism group of ${\Delta}(G)$ are presented and their main properties are investigated.

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References

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