University of Jiroft | hamidreza dorbidi

Research

Title	Commutativity Pattern of Finite Non-Abelian p-Groups Determine Their Orders
Type	Article
Keywords	Graph isomorphism; Groups with abelian centralizers; Non-commuting graph; p-group
Researchers	Alireza Abdollahi, Saieed Akbari, hamidreza dorbidi, Hamid Shahverdi

Abstract

Let G be a non-abelian group and Z(G) be the center of G. Associate a graph \Gamma_G (called noncommuting graph of G) with G as follows: Take G\Z(G) as the vertices of\Gamma_G, and join two distinct vertices x and y, whenever xy is not equal to 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 is isomorphic to \Gamma_H for some group H, then |P| = |H.