Non-nilpotent graph of a group

2009/10/01 by Abdollahi, Alireza, Zarrin, Mohammad · 2 citations
#05C25 #20D60 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.0910.0098

Abstract

We associate a graph N_G with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of N_G and its induced subgraph on G\backslash nil(G), where nil(G)=\x∈ G | < x,y> is nilpotent for all y∈ G\. For any finite group G, we prove that N_G has either |Z^*(G)| or |Z^*(G)|+1 connected components, where Z^*(G) is the hypercenter of G. We give a new characterization for finite nilpotent groups in terms of the non-nilpotent graph. In fact we prove that a finite group G is nilpotent if and only if the set of vertex degrees of N_G has at most two elements.

Cited by

• The nilpotent graph of a finite0-dimensional Lie algebra
• Profinite groups with many elements with large nilpotentizer and generalizations

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