relative n-th non-commuting graphs of finite groups
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abstract
suppose $n$ is a fixed positive integer. we introduce the relative n-th non-commuting graph $gamma^{n} _{h,g}$, associated to the non-abelian subgroup $h$ of group $g$. the vertex set is $gsetminus c^n_{h,g}$ in which $c^n_{h,g} = {xin g : [x,y^{n}]=1 mbox{~and~} [x^{n},y]=1mbox{~for~all~} yin h}$. moreover, ${x,y}$ is an edge if $x$ or $y$ belong to $h$ and $xy^{n}eq y^{n}x$ or $x^{n}yeq yx^{n}$. in fact, the relative n-th commutativity degree, $p_{n}(h,g)$ the probability that n-th power of an element of the subgroup $h$ commutes with another random element of the group $g$ and the non-commuting graph were the keys to construct such a graph. it is proved that two isoclinic non-abelian groups have isomorphic graphs under special conditions.
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Journal title:
bulletin of the iranian mathematical societyPublisher: iranian mathematical society (ims)
ISSN 1017-060X
volume 39
issue 4 2013
Keywords
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