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Paper   IPM / M / 8275
School of Mathematics
  Title:   Non-commuting graph of a group
1.  S. Akbari
2.  H. R. Maimani (Joint with A. Abdollahi)
  Status:   Published
  Journal: J. Algebra
  Vol.:  298
  Year:  2006
  Pages:   468-492
  Supported by:  IPM
Let G be a non-abelian group and let Z(G) be the center of G. Associate a graph ΓG (called non-commuting graph of G) with G as follows: Take G\Z(G) as the vertices of ΓG and join two distinct vertices x and y, whenever xyyx. We want to explore how the graph theoretical properties of ΓG can effect on the group theoretical properties of G. We conjecture that if G and H are two non-abelian finite groups such that ΓG ≅ ΓH, then |G|=|H|. Among other results we show that if G is a finite non-abelian nilpotent group and H is a group such that ΓG ≅ ΓH and |G|=|H|, then H is nilpotent.

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