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

TL;DR

This paper proves that the non-commuting graph of a finite non-abelian p-group uniquely determines its order, establishing a link between the group's commutativity pattern and its size.

Contribution

It demonstrates that the structure of the non-commuting graph uniquely identifies the order of finite non-abelian p-groups, a novel result in group theory.

Findings

Non-commuting graph determines the group's order

Isomorphism of graphs implies equal group orders

Establishes a link between graph structure and group size

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 $xy\neq yx$. 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 $|P|=|H|$.