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 Γ G (called noncommuting graph of G) with G as follows: Take G∖Z(G) as the vertices of Γ G , and join two distinct vertices x and y, whenever xy ≠ 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 Γ P ≅ Γ H for some group H, then |P| = |H|.