Non-commuting graphs of nilpotent groups

2013/04/17 by Alireza Abdollahi, Abdollahi, Alireza, Hamid Shahverdi +1
Mathematics · #20D15 #20D60 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20D15 #msc:20D60

paper · pdf · doi:10.48550/arxiv.1304.4839

5 pages; to appear in Comm. Algebra

arxiv created 2013/04/17 · arxiv updated 2013/04/18

Abstract

Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph Γ_G associated to G is the graph whose vertex set is G∖ Z(G) and two distinct elements x,y are adjacent if and only if xy≠ yx. We prove that if G and H are non-abelian nilpotent groups with irregular isomorphic non-commuting graphs, then |G|=|H|.

Related

• Proof of a conjecture of Abdollahi-Akbari-Maimani concerning the non-commutative graph of finite groups
• A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs
• Non-nilpotent graph of a group
• Commutativity pattern of finite non-abelian p-groups determine their orders
• g-noncommuting graph of a finite group relative to its subgroups