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

2011/09/23 by Aliréza Abdollahi, A. Abdollahi, S. Akbari +9
Engineering · Mathematics · #Finite Group Theory Research #Rings, Modules, and Algebras #graph theory and CDMA systems #math.CO #math.GR #msc:20D15 #msc:20D45

paper · pdf · doi:10.48550/arxiv.1109.5007

to appear in Communications in Algebra

arxiv created 2011/09/23 · arxiv updated 2011/09/26

Abstract

Let G be a non-abelian group and Z(G) be the center of G. Associate a graph Γ_G (called non-commuting 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|.

Related

• Proof of a conjecture of Abdollahi-Akbari-Maimani concerning the non-commutative graph of finite groups
• Non-commuting graphs of nilpotent groups
• On the genus of the commuting graphs of finite non-abelian groups
• Aspects of the commuting graph
• The commuting graphs of certain cyclic-by-abelian groups