“School of Mathematics”
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Paper IPM / M / 8275  


Abstract:  
Let G be a nonabelian group and let Z(G) be the center of
G. Associate a graph Γ_{G} (called noncommuting graph of
G) with G as follows: Take G\Z(G) as the vertices
of Γ_{G} and join two distinct vertices x and y,
whenever xy ≠ yx. We want to explore how the graph theoretical
properties of Γ_{G} can effect on the group theoretical
properties of G. We conjecture that if G and H are two
nonabelian finite groups such that Γ_{G} ≅ Γ_{H},
then G=H. Among other results we show that if G is a
finite nonabelian nilpotent group and H is a group such that
Γ_{G} ≅ Γ_{H} and G=H, then H is
nilpotent.
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