Commutativity pattern of finite non-abelian $p$-groups determine their orders

Abstract

Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. Associate a graph ${\Gamma }_G$ (called non-commuting graph of $G$) with $G$ as follows: take $G\setminus Z(G)$ as the vertices of ${\Gamma }_G$ and join two distinct vertices $x$ and $y$, whenever $xy\ne yx$. Here, we prove that "the commutativity pattern of a finite non-abelian $p$-group determine its order among the class of groups"; this means that if $P$ is a finite non-abelian $p$-group such that ${\Gamma }_P\cong {\Gamma }_H$ for some group $H$, then $|P|=|H|$.

arXiv commentto appear in Communications in Algebra