“School of Mathematics”
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| Paper IPM / M / 8275 |
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| Abstract: | |||||
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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 xy ≠ yx. 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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