Non-solvable graphs of groups

2019/09/26 by Bhowal, Parthajit, Nongsiang, Deiborlang, Nath, Rajat Kanti · 1 citation
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1909.12043

Abstract

Let G be a group and Sol(G)=\x ∈ G : ⟨ x,y ⟩ is solvable for all y ∈ G\. We associate a graph NS_G (called the non-solvable graph of G) with G whose vertex set is G ∖ Sol(G) and two distinct vertices are adjacent if they generate a non-solvable subgroup. In this paper we study many properties of NS_G. In particular, we obtain results on vertex degree, cardinality of vertex degree set, graph realization, domination number, vertex connectivity, independence number and clique number of NS_G. We also consider two groups G and H having isomorphic non-solvable graphs and derive some properties of G and H. Finally, we conclude this paper by showing that NS_G is neither planar, toroidal, double-toroidal, triple-toroidal nor projective.

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